From fdd725e9afda1fb6e1fff64ca4d5dc2d67b9961c Mon Sep 17 00:00:00 2001 From: Trupti Kini Date: Mon, 4 Jan 2016 23:30:16 +0600 Subject: Added(A)/Deleted(D) following books A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/Chapter9_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter1_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter2_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter3_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter4_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter6_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter7_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter8_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter_5_3.ipynb A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex1.2.png A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex3.7.png A Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex6.7.png A Semiconductor_Devices_Basic_Principle_by_J._Singh/Chapter10_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter11_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter1_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter2_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter3_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter5_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter6_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter7_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter8_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter9_1.ipynb A Semiconductor_Devices_Basic_Principle_by_J._Singh/screenshots/ch-1.png A 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Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter_5_3.ipynb create mode 100644 Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex1.2.png create mode 100644 Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex3.7.png create mode 100644 Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/screenshots/Ex6.7.png create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/Chapter10_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter11_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter1_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter2_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter3_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter5_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter6_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter7_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter8_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/chapter9_1.ipynb create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/screenshots/ch-1.png create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/screenshots/ch-10.png create mode 100644 Semiconductor_Devices_Basic_Principle_by_J._Singh/screenshots/ch-6.png diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/Chapter9_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/Chapter9_3.ipynb new file mode 100644 index 00000000..d1825266 --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/Chapter9_3.ipynb @@ -0,0 +1,397 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:6d953e27719d7daa72fde544d6031f1b10e1023af731eda189a4bce609e51019" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter09:Numerical Solution of Partial Differential Equations" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.1:pg-350" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#standard five point formula\n", + "#example 9.1\n", + "#page 350\n", + "\n", + "u2=5.0;u3=1.0;\n", + "for i in range(0,3):\n", + " u1=(u2+u3+6.0)/4.0\n", + " u2=(u1/2.0)+(5.0/2.0)\n", + " u3=(u1/2.0)+(1.0/2.0)\n", + " print\" the values are u1=%d\\t u2=%d\\t u3=%d\\t\\n\\n\" %(u1,u2,u3)\n", + " \n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " the values are u1=3\t u2=4\t u3=2\t\n", + "\n", + "\n", + " the values are u1=3\t u2=4\t u3=2\t\n", + "\n", + "\n", + " the values are u1=3\t u2=4\t u3=2\t\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 30 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.2:pg-351" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#solution of laplace equation by jacobi method,gauss-seidel method and SOR method\n", + "#example 9.2\n", + "#page 351\n", + "u1=0.25;u2=0.25;u3=0.5;u4=0.5;#initial values\n", + "print \"jacobis iteration process\\n\\n\"\n", + "print\"u1\\t u2\\t u3\\t u4\\t \\n\\n\"\n", + "print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4)\n", + "for i in range(0,7):\n", + " u11=(0+u2+0+u4)/4\n", + " u22=(u1+0+0+u3)/4\n", + " u33=(1+u2+0+u4)/4\n", + " u44=(1+0+u3+u1)/4\n", + " u1=u11;u2=u22;u3=u33;u4=u44;\n", + " print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u11,u22,u33,u44) \n", + "print \" gauss seidel process\\n\\n\"\n", + "u1=0.25;u2=0.3125;u3=0.5625;u4=0.46875;#initial values\n", + "print \"u1\\t u2\\t u3\\t u4\\t \\n\\n\"\n", + "print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4)\n", + "for i in range(0,4):\n", + "\n", + " u1=(0.0+u2+0.0+u4)/4.0\n", + " u2=(u1+0.0+0.0+u3)/4.0\n", + " u3=(1.0+u2+0.0+u4)/4.0\n", + " u4=(1.0+0.0+u3+u1)/4.0\n", + " print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4) \n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "jacobis iteration process\n", + "\n", + "\n", + "u1\t u2\t u3\t u4\t \n", + "\n", + "\n", + "0.250000\t 0.250000\t 0.500000\t 0.500000\t \n", + "\n", + "0.187500\t 0.187500\t 0.437500\t 0.437500\t \n", + "\n", + "0.156250\t 0.156250\t 0.406250\t 0.406250\t \n", + "\n", + "0.140625\t 0.140625\t 0.390625\t 0.390625\t \n", + "\n", + "0.132812\t 0.132812\t 0.382812\t 0.382812\t \n", + "\n", + "0.128906\t 0.128906\t 0.378906\t 0.378906\t \n", + "\n", + "0.126953\t 0.126953\t 0.376953\t 0.376953\t \n", + "\n", + "0.125977\t 0.125977\t 0.375977\t 0.375977\t \n", + "\n", + " gauss seidel process\n", + "\n", + "\n", + "u1\t u2\t u3\t u4\t \n", + "\n", + "\n", + "0.250000\t 0.312500\t 0.562500\t 0.468750\t \n", + "\n", + "0.195312\t 0.189453\t 0.414551\t 0.402466\t \n", + "\n", + "0.147980\t 0.140633\t 0.385775\t 0.383439\t \n", + "\n", + "0.131018\t 0.129198\t 0.378159\t 0.377294\t \n", + "\n", + "0.126623\t 0.126196\t 0.375872\t 0.375624\t \n", + "\n" + ] + } + ], + "prompt_number": 51 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.4:pg-354" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#poisson equation\n", + "#exaample 9.4\n", + "#page 354\n", + "u2=0.0;u4=0.0;\n", + "print \" u1\\t u2\\t u3\\t u4\\t\\n\\n\"\n", + "for i in range(0,6):\n", + " u1=(u2/2.0)+30.0\n", + " u2=(u1+u4+150.0)/4.0\n", + " u4=(u2/2.0)+45.0\n", + " print \"%0.2f\\t %0.2f\\t %0.2f\\t %0.2f\\n\" %(u1,u2,u2,u4)\n", + "print \" from last two iterates we conclude u1=67 u2=75 u3=75 u4=83\\n\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " u1\t u2\t u3\t u4\t\n", + "\n", + "\n", + "30.00\t 45.00\t 45.00\t 67.50\n", + "\n", + "52.50\t 67.50\t 67.50\t 78.75\n", + "\n", + "63.75\t 73.12\t 73.12\t 81.56\n", + "\n", + "66.56\t 74.53\t 74.53\t 82.27\n", + "\n", + "67.27\t 74.88\t 74.88\t 82.44\n", + "\n", + "67.44\t 74.97\t 74.97\t 82.49\n", + "\n", + " from last two iterates we conclude u1=67 u2=75 u3=75 u4=83\n", + "\n" + ] + } + ], + "prompt_number": 59 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.6:pg-362" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#bender-schmidt formula\n", + "#example 9.6\n", + "#page 362\n", + "def f(x):\n", + " return (4*x)-(x*x)\n", + "#u=[f(0),f(1),f(2),f(3),f(4)]\n", + "u1=f(0);u2=f(1);u3=f(2);u4=f(3);u5=f(4);\n", + "u11=(u1+u3)/2\n", + "u12=(u2+u4)/2\n", + "u13=(u3+u5)/2\n", + "print \"u11=%0.2f\\t u12=%0.2f\\t u13=%0.2f\\t \\n\" %(u11,u12,u13)\n", + "u21=(u1+u12)/2.0\n", + "u22=(u11+u13)/2.0\n", + "u23=(u12+0)/2.0\n", + "print \"u21=%0.2f\\t u22=%0.2f\\t u23=%0.2f\\t \\n\" %(u21,u22,u23)\n", + "u31=(u1+u22)/2.0\n", + "u32=(u21+u23)/2.0\n", + "u33=(u22+u1)/2.0\n", + "print \"u31=%0.2f\\t u32=%0.2f\\t u33=%0.2f\\t \\n\" % (u31,u32,u33)\n", + "u41=(u1+u32)/2.0\n", + "u42=(u31+u33)/2.0\n", + "u43=(u32+u1)/2.0\n", + "print \"u41=%0.2f\\t u42=%0.2f\\t u43=%0.2f\\t \\n\" % (u41,u42,u43)\n", + "u51=(u1+u42)/2.0\n", + "u52=(u41+u43)/2.0\n", + "u53=(u42+u1)/2.0\n", + "print \"u51=%0.2f\\t u52=%0.2f\\t u53=%0.2f\\t \\n\" % (u51,u52,u53)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "u11=2.00\t u12=3.00\t u13=2.00\t \n", + "\n", + "u21=1.50\t u22=2.00\t u23=1.50\t \n", + "\n", + "u31=1.00\t u32=1.50\t u33=1.00\t \n", + "\n", + "u41=0.75\t u42=1.00\t u43=0.75\t \n", + "\n", + "u51=0.50\t u52=0.75\t u53=0.50\t \n", + "\n" + ] + } + ], + "prompt_number": 77 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.7:pg-363" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "#bender-schimdt's formula and crank-nicolson formula\n", + "#example 9.7\n", + "#page 363\n", + "#bender -schimdt's formula\n", + "import math\n", + "from numpy import matrix\n", + "z=math.pi\n", + "def f(x,t):\n", + " return math.exp(z*z*t*-1)*sin(z*x)\n", + "#u=[f(0,0),f(0.2,0),f(0.4,0),f(0.6,0),f(0.8,0),f(1,0)];\n", + "u1=f(0,0)\n", + "u2=f(0.2,0)\n", + "u3=f(0.4,0)\n", + "u4=f(0.6,0)\n", + "u5=f(0.8,0)\n", + "u6=f(1.0,0)\n", + "u11=u3/2;u12=(u2+u4)/2;u13=u12;u14=u11;\n", + "print \"u11=%f\\t u12=%f\\t u13=%f\\t u14=%f\\n\\n\" % (u11,u12,u13,u14)\n", + "u21=u12/2;u22=(u12+u14)/2;u23=u22;u24=u21;\n", + "print \"u21=%f\\t u22=%f\\t u23=%f\\t u24=%f\\n\\n\" % (u21,u22,u23,u24)\n", + "print \"the error in the solution is: %f\\n\\n\" % (math.fabs(u22-f(0.6,0.04)))\n", + "#crank-nicolson formula\n", + "#by putting i=1,2,3,4 we obtain four equation\n", + "A=matrix([[4, -1, 0, 0] ,[-1, 4, -1, 0],[0, -1, 4, -1],[0, 0, -1, 4]])\n", + "C=matrix([[0.9510],[1.5388],[1.5388],[0.9510]])\n", + "X=A.I*C\n", + "print \"u00=%f\\t u10=%f\\t u20=%f\\t u30=%f\\t\\n\\n\" %(X[0][0],X[1][0],X[2][0],X[3][0])\n", + "print \"the error in the solution is: %f\\n\\n\" %(abs(X[1][0]-f(0.6,0.04)))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "u11=0.475528\t u12=0.769421\t u13=0.769421\t u14=0.475528\n", + "\n", + "\n", + "u21=0.384710\t u22=0.622475\t u23=0.622475\t u24=0.384710\n", + "\n", + "\n", + "the error in the solution is: 0.018372\n", + "\n", + "\n", + "u00=0.399255\t u10=0.646018\t u20=0.646018\t u30=0.399255\t\n", + "\n", + "\n", + "the error in the solution is: 0.005172\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 28 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex9.8:pg-364" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "\n", + "#heat equation using crank-nicolson method\n", + "#example 9.8\n", + "#page 364\n", + "from numpy import matrix\n", + "import math\n", + "z=0.01878;\n", + "#h=1/2;l=1/8,i=1\n", + "u01=0.0;u21=1.0/8.0;\n", + "u11=(u21+u01)/6.0;\n", + "print \" u11=%f\\n\\n\" % (u11)\n", + "print \"error is %f\\n\\n\" % (math.fabs(u11-z))\n", + "#h=1/4,l=1/8,i=1,2,3\n", + "A=matrix([[-3.0 ,-1.0 ,0.0],[1.0,-3.0,1.0],[0.0,1.0,-3.0]])\n", + "C=matrix([[0.0],[0.0],[-0.125]])\n", + "#here we found inverese of A then we multipy it with C\n", + "X=A.I*C\n", + "print \"u12=%f\\n\\n\" % (X[1][0])\n", + "print \"error is %f\\n\\n\" %(math.fabs(X[1][0]-z))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " u11=0.020833\n", + "\n", + "\n", + "error is 0.002053\n", + "\n", + "\n", + "u12=0.013889\n", + "\n", + "\n", + "error is 0.004891\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 23 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter1_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter1_3.ipynb new file mode 100644 index 00000000..cdfe1170 --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter1_3.ipynb @@ -0,0 +1,625 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:ab07d04dc98e9e897aaecd3d260ae7ca8f8d0636f3903bebd7713e9a345cdce9" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter01:Errors in Numerical Calculations" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex1.1:pg-7" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 1.1\n", + "#rounding off\n", + "#page 7\n", + "a1=1.6583\n", + "a2=30.0567\n", + "a3=0.859378\n", + "a4=3.14159\n", + "print \"\\nthe numbers after rounding to 4 significant figures are given below\\n\"\n", + "print \" %f %.4g\\n'\" %(a1,a1)\n", + "print \" %f %.4g\\n\" %(a2,a2)\n", + "print \" %f %.4g\\n\" %(a3,a3)\n", + "print \" %f %.4g\\n\" %(a4,a4)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + "the numbers after rounding to 4 significant figures are given below\n", + "\n", + " 1.658300 1.658\n", + "'\n", + " 30.056700 30.06\n", + "\n", + " 0.859378 0.8594\n", + "\n", + " 3.141590 3.142\n", + "\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex1.2:pg-9" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 1.2\n", + "#percentage accuracy\n", + "#page 9\n", + "import math\n", + "x=0.51 # the number given\n", + "n=2 #correcting upto 2 decimal places\n", + "d=math.pow(10,-n)\n", + "d=d/2.0\n", + "p=(d/x)*100 #percentage accuracy\n", + "print \"the percentage accuracy of %f after correcting to two decimal places is %f\" %(x,p)\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the percentage accuracy of 0.510000 after correcting to two decimal places is 0.980392\n" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex1.3:pg-9" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 1.3\n", + "#absolute and relative errors\n", + "#page 9\n", + "X=3.1428571 #approximate value of pi\n", + "T_X=3.1415926 # true value of pi\n", + "A_E=T_X-X #absolute error\n", + "R_E=A_E/T_X #relative error\n", + "print \"Absolute Error = %0.7f \\n Relative Error = %0.7f\" %(A_E,R_E)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Absolute Error = -0.0012645 \n", + " Relative Error = -0.0004025\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex1.4:pg-10" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 1.4\n", + "#best approximation\n", + "#page 10\n", + "X=1/3 #the actual number\n", + "X1=0.30\n", + "X2=0.33\n", + "X3=0.34\n", + "E1=abs(X-X1)\n", + "E2=abs(X-X2)\n", + "E3=abs(X-X3)\n", + "if E1d:\n", + " m=(x1+x2)/2.0\n", + " print \" \\t%f\\t%f\\t%f\\t%f\\n\" %(x1,x2,m,f(m))\n", + " if f(m)*f(x1)>0:\n", + " x1=m\n", + " else:\n", + " x2=m \n", + " c=c+1;# to count number of iterations \n", + "print \"the solution of equation after %i iteration is %0.4g\" %(c,m)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Succesive approximations \t x1\t \tx2\t \tm\t \tf(m)\n", + "\n", + " \t2.000000\t3.000000\t2.500000\t5.625000\n", + "\n", + " \t2.000000\t2.500000\t2.250000\t1.890625\n", + "\n", + " \t2.000000\t2.250000\t2.125000\t0.345703\n", + "\n", + " \t2.000000\t2.125000\t2.062500\t-0.351318\n", + "\n", + " \t2.062500\t2.125000\t2.093750\t-0.008942\n", + "\n", + " \t2.093750\t2.125000\t2.109375\t0.166836\n", + "\n", + " \t2.093750\t2.109375\t2.101562\t0.078562\n", + "\n", + " \t2.093750\t2.101562\t2.097656\t0.034714\n", + "\n", + " \t2.093750\t2.097656\t2.095703\t0.012862\n", + "\n", + " \t2.093750\t2.095703\t2.094727\t0.001954\n", + "\n", + " \t2.093750\t2.094727\t2.094238\t-0.003495\n", + "\n", + " \t2.094238\t2.094727\t2.094482\t-0.000771\n", + "\n", + " \t2.094482\t2.094727\t2.094604\t0.000592\n", + "\n", + " \t2.094482\t2.094604\t2.094543\t-0.000090\n", + "\n", + "the solution of equation after 15 iteration is 2.095\n" + ] + } + ], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.3:pg-26" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.3\n", + "#bisection method\n", + "#page 26\n", + "import math\n", + "def f(x):\n", + " return math.pow(x,3)+math.pow(x,2)+x+7\n", + "x1=-3\n", + "x2=-2 #f(-3) is negative and f(-2) is positive\n", + "d=0.0001 #for accuracy of root\n", + "c=1\n", + "print \"Succesive approximations \\t x1\\t \\tx2\\t \\tm\\t \\tf(m)\\n\"\n", + "while abs(x1-x2)>d:\n", + " m=(x1+x2)/2.0\n", + " print \" \\t%f\\t%f\\t%f\\t%f\\n\" %(x1,x2,m,f(m))\n", + " if f(m)*f(x1)>0:\n", + " x1=m\n", + " else:\n", + " x2=m \n", + " c=c+1 # to count number of iterations \n", + "print \"the solution of equation after %i iteration is %0.4g\" %(c,m)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Succesive approximations \t x1\t \tx2\t \tm\t \tf(m)\n", + "\n", + " \t-3.000000\t-2.000000\t-2.500000\t-4.875000\n", + "\n", + " \t-2.500000\t-2.000000\t-2.250000\t-1.578125\n", + "\n", + " \t-2.250000\t-2.000000\t-2.125000\t-0.205078\n", + "\n", + " \t-2.125000\t-2.000000\t-2.062500\t0.417725\n", + "\n", + " \t-2.125000\t-2.062500\t-2.093750\t0.111481\n", + "\n", + " \t-2.125000\t-2.093750\t-2.109375\t-0.045498\n", + "\n", + " \t-2.109375\t-2.093750\t-2.101562\t0.033315\n", + "\n", + " \t-2.109375\t-2.101562\t-2.105469\t-0.006010\n", + "\n", + " \t-2.105469\t-2.101562\t-2.103516\t0.013673\n", + "\n", + " \t-2.105469\t-2.103516\t-2.104492\t0.003836\n", + "\n", + " \t-2.105469\t-2.104492\t-2.104980\t-0.001086\n", + "\n", + " \t-2.104980\t-2.104492\t-2.104736\t0.001376\n", + "\n", + " \t-2.104980\t-2.104736\t-2.104858\t0.000145\n", + "\n", + " \t-2.104980\t-2.104858\t-2.104919\t-0.000470\n", + "\n", + "the solution of equation after 15 iteration is -2.105\n" + ] + } + ], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.4:pg-26" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.4\n", + "#bisection method\n", + "#page 26\n", + "import math\n", + "def f(x):\n", + " return x*math.exp(x)-1\n", + "x1=0 \n", + "x2=1 #f(0) is negative and f(1) is positive\n", + "d=0.0005 #maximun tolerance value\n", + "c=1\n", + "print \"Succesive approximations \\t x1\\t \\tx2\\t \\tm\\t \\ttol\\t \\tf(m)\\n\"\n", + "while abs((x2-x1)/x2)>d:\n", + " m=(x1+x2)/2.0 #tolerance value for each iteration\n", + " tol=((x2-x1)/x2)*100\n", + " print \" \\t%f\\t%f\\t%f\\t%f\\t%f\\n\" %(x1,x2,m,tol,f(m))\n", + " if f(m)*f(x1)>0:\n", + " x1=m\n", + " else:\n", + " x2=m \n", + " c=c+1 # to count number of iterations \n", + "print \"the solution of equation after %i iteration is %0.4g\" %(c,m)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Succesive approximations \t x1\t \tx2\t \tm\t \ttol\t \tf(m)\n", + "\n", + " \t0.000000\t1.000000\t0.500000\t100.000000\t-0.175639\n", + "\n", + " \t0.500000\t1.000000\t0.750000\t50.000000\t0.587750\n", + "\n", + " \t0.500000\t0.750000\t0.625000\t33.333333\t0.167654\n", + "\n", + " \t0.500000\t0.625000\t0.562500\t20.000000\t-0.012782\n", + "\n", + " \t0.562500\t0.625000\t0.593750\t10.000000\t0.075142\n", + "\n", + " \t0.562500\t0.593750\t0.578125\t5.263158\t0.030619\n", + "\n", + " \t0.562500\t0.578125\t0.570312\t2.702703\t0.008780\n", + "\n", + " \t0.562500\t0.570312\t0.566406\t1.369863\t-0.002035\n", + "\n", + " \t0.566406\t0.570312\t0.568359\t0.684932\t0.003364\n", + "\n", + " \t0.566406\t0.568359\t0.567383\t0.343643\t0.000662\n", + "\n", + " \t0.566406\t0.567383\t0.566895\t0.172117\t-0.000687\n", + "\n", + " \t0.566895\t0.567383\t0.567139\t0.086059\t-0.000013\n", + "\n", + "the solution of equation after 13 iteration is 0.5671\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.5:pg-27" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.5\n", + "#bisection method\n", + "#page 27\n", + "import math\n", + "def f(x):\n", + " return 4*math.exp(-x)*math.sin(x)-1\n", + "x1=0 \n", + "x2=0.5 #f(0) is negative and f(1) is positive\n", + "d=0.0001 #for accuracy of root\n", + "c=1 \n", + "print \"Succesive approximations \\t x1\\t \\tx2\\t \\tm\\t \\t \\tf(m)\\n\"\n", + "while abs(x2-x1)>d:\n", + " m=(x1+x2)/2.0\n", + " print \" \\t%f\\t%f\\t%f\\t%f\\n\" %(x1,x2,m,f(m))\n", + " if f(m)*f(x1)>0:\n", + " x1=m\n", + " else:\n", + " x2=m \n", + " c=c+1 # to count number of iterations \n", + "print \"the solution of equation after %i iteration is %0.3g\" %(c,m)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "Succesive approximations \t x1\t \tx2\t \tm\t \t \tf(m)\n", + "\n", + " \t0.000000\t0.500000\t0.250000\t-0.229286\n", + "\n", + " \t0.250000\t0.500000\t0.375000\t0.006941\n", + "\n", + " \t0.250000\t0.375000\t0.312500\t-0.100293\n", + "\n", + " \t0.312500\t0.375000\t0.343750\t-0.044068\n", + "\n", + " \t0.343750\t0.375000\t0.359375\t-0.017925\n", + "\n", + " \t0.359375\t0.375000\t0.367188\t-0.005334\n", + "\n", + " \t0.367188\t0.375000\t0.371094\t0.000842\n", + "\n", + " \t0.367188\t0.371094\t0.369141\t-0.002236\n", + "\n", + " \t0.369141\t0.371094\t0.370117\t-0.000694\n", + "\n", + " \t0.370117\t0.371094\t0.370605\t0.000075\n", + "\n", + " \t0.370117\t0.370605\t0.370361\t-0.000310\n", + "\n", + " \t0.370361\t0.370605\t0.370483\t-0.000118\n", + "\n", + " \t0.370483\t0.370605\t0.370544\t-0.000022\n", + "\n", + "the solution of equation after 14 iteration is 0.371\n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.6:pg-28" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.6\n", + "#false position method\n", + "#page 28\n", + "import math\n", + "def f(x):\n", + " return x**3-2*x-5\n", + "a=2.0\n", + "b=3.0 #f(2) is negative and f(3)is positive\n", + "d=0.00001\n", + "print \"succesive iterations \\ta\\t b\\t f(a)\\t f(b)\\t\\ x1\\n\"\n", + "for i in range(1,25):\n", + " x1=b*f(a)/(f(a)-f(b))+a*f(b)/(f(b)-f(a))\n", + " if(f(a)*f(x1))>0:\n", + " b=x1\n", + " else:\n", + " a=x1\n", + " if abs(f(x1))0:\n", + " b=x1\n", + " else:\n", + " a=x1\n", + " if abs(f(x1))0:\n", + " b=x1\n", + " else:\n", + " a=x1\n", + " if abs(f(x1))0:\n", + " b=x1\n", + " else:\n", + " a=x1\n", + " if abs(f(x1))d:\n", + " print \" \\t%f %f\\n\" %(x1,f(x1))\n", + " x2=x1\n", + " x1=f(x1)\n", + " c=c+1\n", + "print \" the root of the eqaution after %i iteration is %0.4g\" %(c,x1)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t\u0001\tf(x1)\n", + "\n", + " \t0.750000 0.755929\n", + "\n", + " \t0.755929 0.754652\n", + "\n", + " \t0.754652 0.754926\n", + "\n", + " \t0.754926 0.754867\n", + "\n", + " the root of the eqaution after 4 iteration is 0.7549\n" + ] + } + ], + "prompt_number": 23 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.11:pg-34" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.11\n", + "#iteration method\n", + "#page34\n", + "import math\n", + "def f(x):\n", + " return cos(x)/2.0+3.0/2.0\n", + "x1=1.5 # as roots lies between 3/2 and pi/2\n", + "x2=0\n", + "d=0.0001 # accuracy opto 10^-4\n", + "c=0 # to count no of iterations \n", + "print \"successive iterations \\t\\x01\\tf(x1)\\n\"\n", + "while abs(x2-x1)>d:\n", + " \n", + " print \" \\t%f %f\\n\" %(x1,f(x1))\n", + " x2=x1\n", + " x1=f(x1)\n", + " c=c+1\n", + "print \" the root of the eqaution after %i iteration is %0.4g\" %(c,x1)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t\u0001\tf(x1)\n", + "\n", + " \t1.500000 1.535369\n", + "\n", + " \t1.535369 1.517710\n", + "\n", + " \t1.517710 1.526531\n", + "\n", + " \t1.526531 1.522126\n", + "\n", + " \t1.522126 1.524326\n", + "\n", + " \t1.524326 1.523227\n", + "\n", + " \t1.523227 1.523776\n", + "\n", + " \t1.523776 1.523502\n", + "\n", + " \t1.523502 1.523639\n", + "\n", + " \t1.523639 1.523570\n", + "\n", + " the root of the eqaution after 10 iteration is 1.524\n" + ] + } + ], + "prompt_number": 28 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.12:pg-35" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.12\n", + "#iteration method\n", + "#page 35\n", + "import math\n", + "def f(x):\n", + " return math.exp(-x)\n", + "x1=1.5 # as roots lies between 0 and 1\n", + "x2=0\n", + "d=0.0001 # accuracy opto 10^-4\n", + "c=0 # to count no of iterations \n", + "print \"successive iterations \\t x1 \\t f(x1)\\n\"\n", + "while abs(x2-x1)>d:\n", + " \n", + " print \" \\t%f %f\\n\" %(x1,f(x1))\n", + " x2=x1\n", + " x1=f(x1)\n", + " c=c+1\n", + "print \" the root of the eqaution after %i iteration is %0.4g\" %(c,x1)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t x1 \t f(x1)\n", + "\n", + " \t1.500000 0.223130\n", + "\n", + " \t0.223130 0.800011\n", + "\n", + " \t0.800011 0.449324\n", + "\n", + " \t0.449324 0.638059\n", + "\n", + " \t0.638059 0.528317\n", + "\n", + " \t0.528317 0.589597\n", + "\n", + " \t0.589597 0.554551\n", + "\n", + " \t0.554551 0.574330\n", + "\n", + " \t0.574330 0.563082\n", + "\n", + " \t0.563082 0.569451\n", + "\n", + " \t0.569451 0.565836\n", + "\n", + " \t0.565836 0.567885\n", + "\n", + " \t0.567885 0.566723\n", + "\n", + " \t0.566723 0.567382\n", + "\n", + " \t0.567382 0.567008\n", + "\n", + " \t0.567008 0.567220\n", + "\n", + " \t0.567220 0.567100\n", + "\n", + " \t0.567100 0.567168\n", + "\n", + " the root of the eqaution after 18 iteration is 0.5672\n" + ] + } + ], + "prompt_number": 34 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.13:pg-35" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.13\n", + "#iteration method\n", + "#page 35\n", + "import math\n", + "def f(x):\n", + " return 1+math.sin(x)/10\n", + "x1=1.0 # as roots lies between 1 and pi evident from graph\n", + "x2=0\n", + "d=0.0001 # accuracy opto 10^-4\n", + "c=0 # to count no of iterations \n", + "print \"successive iterations \\t x1 \\t f(x1)\\n\"\n", + "while abs(x2-x1)>d:\n", + " print \" \\t%f %f\\n\" %(x1,f(x1))\n", + " x2=x1\n", + " x1=f(x1)\n", + " c=c+1\n", + "print \" the root of the eqaution after %i iteration is %0.4g\" %(c,x1)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t x1 \t f(x1)\n", + "\n", + " \t1.000000 1.084147\n", + "\n", + " \t1.084147 1.088390\n", + "\n", + " \t1.088390 1.088588\n", + "\n", + " \t1.088588 1.088597\n", + "\n", + " the root of the eqaution after 4 iteration is 1.089\n" + ] + } + ], + "prompt_number": 36 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.14:pg-36" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.14\n", + "#aitken's process\n", + "#page 36\n", + "import math\n", + "def f(x):\n", + " return 1.5+math.cos(x)/2.0\n", + "x0=1.5\n", + "y=0\n", + "e=0.0001\n", + "c=0\n", + "print \"successive iterations \\t x0 \\t x1 \\t x2 \\t x3 \\t y\\n\"\n", + "for i in range(1,10):\n", + " x1=f(x0)\n", + " x2=f(x1)\n", + " x3=f(x2)\n", + " y=x3-((x3-x2)**2)/(x3-2*x2+x1)\n", + " d=y-x0\n", + " x0=y\n", + " if abs(f(x0))0:\n", + " x2=x3;\n", + " else:\n", + " x1=x3 \n", + " if abs(f(x3))<0.000001: \n", + " break\n", + " c=c+1\n", + "print \"the root of the equation after %i iteration is: %f\" %(c,x3)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t x1 \t x2 \t x3 \t f(x3)\n", + "\n", + " \t2.000000 \t3.000000 \t2.058824 \t-0.390800\n", + "\n", + " \t2.000000 \t2.058824 \t2.096559 \t0.022428\n", + "\n", + " \t2.096559 \t2.058824 \t2.094511 \t-0.000457\n", + "\n", + " \t2.094511 \t2.058824 \t2.094552 \t0.000009\n", + "\n", + " \t2.094552 \t2.058824 \t2.094551 \t-0.000000\n", + "\n", + "the root of the equation after 4 iteration is: 2.094551\n" + ] + } + ], + "prompt_number": 29 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.26:pg-50" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 2.26\n", + "#secant method\n", + "#page 50\n", + "import math\n", + "from __future__ import division\n", + "def f(x):\n", + " return x*math.exp(x)-1\n", + "x1=0\n", + "x2=1 # initial values\n", + "n=1\n", + "c=0 \n", + "print \"successive iterations \\t x1 \\t x2 \\t x3 \\t f(x3)\\n\"\n", + "while n==1:\n", + " x3=(x1*f(x2)-x2*f(x1))/(f(x2)-f(x1)) \n", + " print \" \\t%f \\t%f \\t%f \\t%f\\n\" %(x1,x2,x3,f(x3))\n", + " if f(x3)*f(x1)>0:\n", + " x2=x3\n", + " else:\n", + " x1=x3 \n", + " if abs(f(x3))<0.0001:\n", + " break\n", + " c=c+1\n", + "print \"the root of the equation after %i iteration is: %0.4g\" %(c,x3)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t x1 \t x2 \t x3 \t f(x3)\n", + "\n", + " \t0.000000 \t1.000000 \t0.367879 \t-0.468536\n", + "\n", + " \t0.000000 \t0.367879 \t0.692201 \t0.383091\n", + "\n", + " \t0.692201 \t0.367879 \t0.546310 \t-0.056595\n", + "\n", + " \t0.546310 \t0.367879 \t0.570823 \t0.010200\n", + "\n", + " \t0.570823 \t0.367879 \t0.566500 \t-0.001778\n", + "\n", + " \t0.566500 \t0.367879 \t0.567256 \t0.000312\n", + "\n", + " \t0.567256 \t0.367879 \t0.567124 \t-0.000055\n", + "\n", + "the root of the equation after 6 iteration is: 0.5671\n" + ] + } + ], + "prompt_number": 31 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.27:pg-52" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# example 2.27\n", + "#mulller's method\n", + "#page 52\n", + "from __future__ import division\n", + "import math\n", + "def f(x):\n", + " return x**3-x-1\n", + "x0=0\n", + "x1=1\n", + "x2=2 # initial values\n", + "n=1\n", + "c=0\n", + "print \"successive iterations \\t x0 \\t x1 \\t x2 \\t f(x0)\\t f(x1)\\t f(x2)\\n\"\n", + "while n==1: \n", + " c=c+1\n", + " y0=f(x0)\n", + " y1=f(x1)\n", + " y2=f(x2)\n", + " h2=x2-x1\n", + " h1=x1-x0\n", + " d2=f(x2)-f(x1)\n", + " d1=f(x1)-f(x0)\n", + " print \" \\t%f\\t %f\\t %f\\t %f\\t %f\\t %f\\n\" %(x0,x1,x2,f(x0),f(x1),f(x2))\n", + " A=(d2/h2-d1/h1)/(h1+h2)\n", + " B=d2/h2+A*h2\n", + " S=math.sqrt(B**2-4*A*f(x2))\n", + " x3=x2-(2*f(x2))/(B+S)\n", + " E=abs((x3-x2)/x2)*100\n", + " if E<0.003:\n", + " break\n", + " else:\n", + " if c==1:\n", + " x2=x3\n", + " if c==2:\n", + " x1=x2\n", + " x2=x3\n", + " if c==3:\n", + " x0=x1\n", + " x1=x2\n", + " x2=x3\n", + " if c==3:\n", + " c=0\n", + "print \"the required root is : %0.4f\" %(x3)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "successive iterations \t x0 \t x1 \t x2 \t f(x0)\t f(x1)\t f(x2)\n", + "\n", + " \t0.000000\t 1.000000\t 2.000000\t -1.000000\t -1.000000\t 5.000000\n", + "\n", + " \t0.000000\t 1.000000\t 1.263763\t -1.000000\t -1.000000\t -0.245412\n", + "\n", + " \t0.000000\t 1.263763\t 1.331711\t -1.000000\t -0.245412\t 0.030015\n", + "\n", + " \t1.263763\t 1.331711\t 1.324583\t -0.245412\t 0.030015\t -0.000574\n", + "\n", + " \t1.263763\t 1.331711\t 1.324718\t -0.245412\t 0.030015\t -0.000000\n", + "\n", + "the required root is : 1.3247\n" + ] + } + ], + "prompt_number": 35 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.28:pg-55" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#graeffe's method\n", + "#example 2.28\n", + "#page 55\n", + "import math\n", + "from __future__ import division\n", + "def f(x):\n", + " return x**3-6*(x**2)+11*x-6\n", + "#x=poly(0,'x')\n", + "#g=f(-x)\n", + "print \"the equation is:\\n\"\n", + "A=[1, 14, 49, 36] #coefficients of the above equation\n", + "print \"%0.4g\\n\" %(math.sqrt(A[3]/A[2]))\n", + "print \"%0.4g\\n\" %(math.sqrt(A[2]/A[1]))\n", + "print \"%0.4g\\n\" %(math.sqrt(A[1]/A[0]))\n", + "print \"the equation is:\\n\"\n", + "#disp(g*(-1*g));\n", + "B=[1, 98, 1393, 1296]\n", + "print \"%0.4g\\n\" %((B[3]/B[2])**(1/4))\n", + "print \"%0.4g\\n\" %((B[2]/B[1])**(1/4))\n", + "print \"%0.4g\\n\" %((B[1]/B[0])**(1/4))\n", + "print \"It is apparent from the outputs that the roots converge at 1 2 3\"\n", + "\n", + "\n", + "\n", + "#INCOMPLETE" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the equation is:\n", + "\n", + "0.8571\n", + "\n", + "1.871\n", + "\n", + "3.742\n", + "\n", + "the equation is:\n", + "\n", + "0.9821\n", + "\n", + "1.942\n", + "\n", + "3.146\n", + "\n", + "It is apparent from the outputs that the roots converge at 1 2 3\n" + ] + } + ], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.29:pg-57" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#quadratic factor by lin's--bairsttow method\n", + "#example 2.29\n", + "#page 57\n", + "from numpy import matrix\n", + "from __future__ import division\n", + "def f(x):\n", + " return x**3-x-1\n", + "a=[-1, -1, 0, 1]\n", + "r1=1\n", + "s1=1\n", + "b4=a[3]\n", + "def f3(r):\n", + " return a[2]-r*a[3]\n", + "def f2(r,s):\n", + " return a[1]-r*a[2]+r**2*a[3]-s*a[3]\n", + "def f1(r,s):\n", + " return a[0]-s*a[2]+s*r*a[3]\n", + "A=matrix([[1,1],[2,-1]])\n", + "C=matrix([[0],[1]])\n", + "X=A.I*C\n", + "X1=[[ 0.33333333],[-0.33333333]]\n", + "dr=X1[0][0]\n", + "ds=X1[1][0]\n", + "r2=r1+dr\n", + "s2=s1+ds\n", + "#second pproximation\n", + "r1=r2\n", + "s1=s2\n", + "b11=f1(r2,s2)\n", + "b22=f2(r2,s2)\n", + "h=0.001\n", + "dr_b1=(f1(r1+h,s1)-f1(r1,s1))/h\n", + "ds_b1=(f1(r1,s1+h)-f1(r1,s1))/h\n", + "dr_b2=(f2(r1+h,s1)-f2(r1,s1))/h\n", + "ds_b2=(f2(r1,s1+h)-f2(r1,s1))/h\n", + "A=matrix([[dr_b1,ds_b1],[dr_b2,ds_b2]])\n", + "C=matrix([[-f1(r1,s1)],[-f2(r1,s2)]])\n", + "X=A.I*C\n", + "r2=r1+X[0][0]\n", + "s2=s1+X[1][0]\n", + "print \"roots correct to 3 decimal places are : %0.3f %0.3f\" %(r2,s2)\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "roots correct to 3 decimal places are : 1.325 0.754\n" + ] + } + ], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.31:pg-62" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#method of iteration\n", + "#example 2.31\n", + "#page 62\n", + "from __future__ import division\n", + "def f(x,y):\n", + " return (3*y*x**2+7)/10\n", + "def g(x,y):\n", + " return (y**2+4)/5\n", + "h=0.0001\n", + "x0=0.5\n", + "y0=0.5\n", + "f1_dx=(f(x0+h,y0)-f(x0,y0))/h\n", + "f1_dy=(f(x0,y0+h)-f(x0,y0))/h\n", + "g1_dx=(g(x0+h,y0)-g(x0,y0))/h\n", + "g1_dy=(g(x0+h,y0)-g(x0,y0))/h\n", + "if (f1_dx+f1_dy<1) and (g1_dx+g1_dy<1): \n", + " print \"coditions for convergence is satisfied\\n\\n\"\n", + "print \"X \\t Y\\t\\n\\n\"\n", + "for i in range(0,10):\n", + " X=(3*y0*x0**2+7)/10\n", + " Y=(y0**2+4)/5\n", + " print \"%f\\t %f\\t\\n\" %(X,Y)\n", + " x0=X\n", + " y0=Y\n", + "print \"\\n\\n CONVERGENCE AT (1 1) IS OBVIOUS\"\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "coditions for convergence is satisfied\n", + "\n", + "\n", + "X \t Y\t\n", + "\n", + "\n", + "0.737500\t 0.850000\t\n", + "\n", + "0.838696\t 0.944500\t\n", + "\n", + "0.899312\t 0.978416\t\n", + "\n", + "0.937391\t 0.991460\t\n", + "\n", + "0.961360\t 0.996598\t\n", + "\n", + "0.976320\t 0.998642\t\n", + "\n", + "0.985572\t 0.999457\t\n", + "\n", + "0.991247\t 0.999783\t\n", + "\n", + "0.994707\t 0.999913\t\n", + "\n", + "0.996807\t 0.999965\t\n", + "\n", + "\n", + "\n", + " CONVERGENCE AT (1 1) IS OBVIOUS\n" + ] + } + ], + "prompt_number": 9 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.32:pg-65" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#newton raphson method\n", + "#example 2.32\n", + "#page 65\n", + "def f(x,y):\n", + " return 3*y*x**2-10*x+7\n", + "def g(y):\n", + " return y**2-5*y+4\n", + "hh=0.0001\n", + "x0=0.5\n", + "y0=0.5 #initial values\n", + "f0=f(x0,y0)\n", + "g0=g(y0)\n", + "df_dx=(f(x0+hh,y0)-f(x0,y0))/hh\n", + "df_dy=(f(x0,y0+hh)-f(x0,y0))/hh\n", + "dg_dx=(g(y0)-g(y0))/hh\n", + "dg_dy=(g(y0+hh)-g(y0))/hh\n", + "d=[[df_dx,df_dy],[dg_dx,dg_dy]]\n", + "D1=det(d)\n", + "dd=[[-f0,df_dy],[-g0,dg_dy]]\n", + "h=det(dd)/D1\n", + "ddd=[[df_dx,-f0],[dg_dx,-g0]]\n", + "k=det(ddd)/D1;\n", + "x1=x0+h\n", + "y1=y0+k\n", + "f0=f(x1,y1)\n", + "g0=g(y1)\n", + "df_dx=(f(x1+hh,y1)-f(x1,y1))/hh\n", + "df_dy=(f(x1,y1+hh)-f(x1,y1))/hh\n", + "dg_dx=(g(y1)-g(y1))/hh\n", + "dg_dy=(g(y1+hh)-g(y1))/hh\n", + "dddd=[[df_dx,df_dy],[dg_dx,dg_dy]]\n", + "D2=det(dddd)\n", + "ddddd=[[-f0,df_dy],[-g0,dg_dy]]\n", + "h=det(ddddd)/D2\n", + "d6=[[df_dx,-f0],[dg_dx,-g0]]\n", + "k=det(d6)/D2\n", + "x2=x1+h\n", + "y2=y1+k\n", + "print \" the roots of the equation are x2=%f and y2=%f\" %(x2,y2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " the roots of the equation are x2=0.970803 and y2=0.998752\n" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.33:pg-66" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#newton raphson method\n", + "#example 2.33\n", + "#page 66\n", + "import math\n", + "def f(x,y):\n", + " return x**2+y**2-1\n", + "def g(x,y):\n", + " return y-x**2\n", + "hh=0.0001\n", + "x0=0.7071\n", + "y0=0.7071 #initial values\n", + "f0=f(x0,y0)\n", + "g0=g(x0,y0)\n", + "df_dx=(f(x0+hh,y0)-f(x0,y0))/hh\n", + "df_dy=(f(x0,y0+hh)-f(x0,y0))/hh\n", + "dg_dx=(g(x0+hh,y0)-g(x0,y0))/hh\n", + "dg_dy=(g(x0,y0+hh)-g(x0,y0))/hh\n", + "D1=det([[df_dx,df_dy],[dg_dx,dg_dy]])\n", + "h=det([[-f0,df_dy],[-g0,dg_dy]])/D1\n", + "k=det([[df_dx,-f0],[dg_dx,-g0]])/D1\n", + "x1=x0+h\n", + "y1=y0+k\n", + "f0=f(x1,y1)\n", + "g0=g(x1,y1)\n", + "df_dx=(f(x1+hh,y1)-f(x1,y1))/hh\n", + "df_dy=(f(x1,y1+hh)-f(x1,y1))/hh\n", + "dg_dx=(g(x1+hh,y1)-g(x1,y1))/hh\n", + "dg_dy=(g(x1,y1+hh)-g(x1,y1))/hh\n", + "D2=det([[df_dx,df_dy],[dg_dx,dg_dy]])\n", + "h=det([[-f0,df_dy],[-g0,dg_dy]])/D2\n", + "k=det([[df_dx,-f0],[dg_dx,-g0]])/D2\n", + "x2=x1+h\n", + "y2=y1+k\n", + "print \"the roots of the equation are x2=%f and y2=%f \" %(x2,y2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the roots of the equation are x2=0.786184 and y2=0.618039 \n" + ] + } + ], + "prompt_number": 11 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.34:pg-67" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#newton raphson method\n", + "#example 2.34\n", + "#page 67\n", + "import math\n", + "def f(x,y):\n", + " return math.sin(x)-y+0.9793\n", + "def g(x,y):\n", + " return math.cos(y)-x+0.6703\n", + "hh=0.0001\n", + "x0=0.5\n", + "y0=1.5 #initial values\n", + "f0=f(x0,y0)\n", + "g0=g(x0,y0)\n", + "df_dx=(f(x0+hh,y0)-f(x0,y0))/hh\n", + "df_dy=(f(x0,y0+hh)-f(x0,y0))/hh\n", + "dg_dx=(g(x0+hh,y0)-g(x0,y0))/hh\n", + "dg_dy=(g(x0,y0+hh)-g(x0,y0))/hh\n", + "d1=[[df_dx,df_dy],[dg_dx,dg_dy]]\n", + "D1=det(d1)\n", + "d2=[[-f0,df_dy],[-g0,dg_dy]]\n", + "h=det(d2)/D1\n", + "d3=[[df_dx,-f0],[dg_dx,-g0]]\n", + "k=det(d3)/D1\n", + "x1=x0+h\n", + "y1=y0+k\n", + "f0=f(x1,y1)\n", + "g0=g(x1,y1)\n", + "df_dx=(f(x1+hh,y1)-f(x1,y1))/hh\n", + "df_dy=(f(x1,y1+hh)-f(x1,y1))/hh\n", + "dg_dx=(g(x1+hh,y1)-g(x1,y1))/hh\n", + "dg_dy=(g(x1,y1+hh)-g(x1,y1))/hh\n", + "d4=[[df_dx,df_dy],[dg_dx,dg_dy]]\n", + "D2=det(d4)\n", + "h=det([[-f0,df_dy],[-g0,dg_dy]])/D2\n", + "k=det([[df_dx,-f0],[dg_dx,-g0]])/D2\n", + "x2=x1+h\n", + "y2=y1+k\n", + "print \"the roots of the equation are x2=%0.4f and y2=%0.4f\" %(x2,y2)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the roots of the equation are x2=0.6537 and y2=1.5874\n" + ] + } + ], + "prompt_number": 9 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter3_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter3_3.ipynb new file mode 100644 index 00000000..77d8f79f --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter3_3.ipynb @@ -0,0 +1,1113 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:58def12f7e424e92e928d020c21b40714eff26275c7ce87aa5600004fbc92a49" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter03:Interpolation" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.4:pg-86" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.4\n", + "#interpolation\n", + "#page 86\n", + "import math\n", + "from __future__ import division\n", + "x=[1, 3, 5, 7]\n", + "y=[24, 120, 336, 720]\n", + "d1=[0,0,0]\n", + "d2=[0,0,0]\n", + "d3=[0,0,0]\n", + "h=2 #interval between values of x\n", + "c=0\n", + "for i in range(0,3):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,2):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,1):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "d=[0,d1[0],d2[0],d3[0]]\n", + "x0=8 #value at 8\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-1)/2\n", + "for i in range(1,4):\n", + " pp=1\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i)\n", + "print \"value of function at %f is :%f\" %(x0,y_x)\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "value of function at 8.000000 is :990.000000\n" + ] + } + ], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.6:pg-87" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.6\n", + "#interpolation\n", + "#page 87\n", + "x=[15, 20, 25, 30, 35, 40]\n", + "y=[0.2588190, 0.3420201, 0.4226183, 0.5, 0.5735764, 0.6427876]\n", + "d1=[0,0,0,0,0]\n", + "d2=[0,0,0,0]\n", + "d3=[0,0,0]\n", + "d4=[0,0]\n", + "d5=[0]\n", + "h=5 #interval between values of x\n", + "c=0\n", + "for i in range(0,5):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,2):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,1):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1\n", + "c=0\n", + "d=[0,d1[0], d2[0], d3[0], d4[0], d5[0]]\n", + "x0=38 #value at 38 degree\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-x[0])/h\n", + "for i in range(1,6):\n", + " pp=1\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+((pp*d[i])/math.factorial(i));\n", + "print \"value of function at %i is :%f\" %(x0,y_x)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "value of function at 38 is :0.615661\n" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.7:pg-89" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.7\n", + "#interpolation\n", + "#page 89\n", + "x=[0, 1, 2, 4]\n", + "y=[1, 3, 9, 81]\n", + "#equation is y(5)-4*y(4)+6*y(2)-4*y(2)+y(1)\n", + "y3=(y[3]+6*y[2]-4*y[1]+y[0])/4\n", + "print \"the value of missing term of table is :%d\" %(y3)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of missing term of table is :31\n" + ] + } + ], + "prompt_number": 1 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.8:pg-89" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.8\n", + "#interpolation\n", + "#page 89\n", + "import math\n", + "x=[0.10, 0.15, 0.20, 0.25, 0.30]\n", + "y=[0.1003, 0.1511, 0.2027, 0.2553, 0.3093]\n", + "d1=[0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0,0]\n", + "d4=[0,0,0,0,0]\n", + "h=0.05 #interval between values of x\n", + "c=0\n", + "for i in range(0,4):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,2):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1\n", + "d=[0,d1[0], d2[0], d3[0], d4[0]]\n", + "x0=0.12 #value at 0.12;\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-x[0])/h\n", + "for i in range(1,5):\n", + " pp=1;\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i)\n", + "print \"value of function at %f is :%0.4g\\n \\n\" %(x0,y_x)\n", + "x0=0.26 #value at 0.26;\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-x[0])/h\n", + "for i in range(1,5):\n", + " pp=1\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i);\n", + "print \"value of function at %f is :%0.4g\\n \\n\" %(x0,y_x)\n", + "x0=0.40 #value at 0.40;\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-x[0])/h\n", + "for i in range(1,5):\n", + " pp=1\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i)\n", + "print \"value of function at %f is :%0.4g\\n \\n\" %(x0,y_x)\n", + "x0=0.50 #value at 0.50;\n", + "pp=1\n", + "y_x=y[0]\n", + "p=(x0-x[0])/h\n", + "for i in range(1,5):\n", + " pp=1\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i)\n", + "print \"value of function at %f is :%0.5g\\n \\n\" %(x0,y_x)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "value of function at 0.120000 is :0.1205\n", + " \n", + "\n", + "value of function at 0.260000 is :0.266\n", + " \n", + "\n", + "value of function at 0.400000 is :0.4241\n", + " \n", + "\n", + "value of function at 0.500000 is :0.5543\n", + " \n", + "\n" + ] + } + ], + "prompt_number": 33 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.9:pg-93" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.9\n", + "#Gauss' forward formula\n", + "#page 93\n", + "x=[1.0, 1.05, 1.10, 1.15, 1.20, 1.25, 1.30];\n", + "y=[2.7183, 2.8577, 3.0042, 3.1582, 3.3201, 3.4903, 3.66693]\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "d5=[0,0]\n", + "d6=[0]\n", + "h=0.05 #interval between values of x\n", + "c=0\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,2):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,1):\n", + " d6[c]=d5[i+1]-d5[i]\n", + " c=c+1\n", + "d=[0,d1[3], d2[2], d3[2], d4[1], d5[0], d6[0]]\n", + "x0=1.17 #value at 1.17;\n", + "pp=1\n", + "y_x=y[3]\n", + "p=(x0-x[3])/h\n", + "for i in range(1,6):\n", + " pp=1;\n", + " for j in range(0,i):\n", + " pp=pp*(p-(j)) \n", + " y_x=y_x+(pp*d[i])/math.factorial(i)\n", + "print \"value of function at %f is :%0.4g\\n \\n\" %(x0,y_x)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "value of function at 1.170000 is :3.222\n", + " \n", + "\n" + ] + } + ], + "prompt_number": 35 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.10:pg-97" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#practical interpolation\n", + "#example 3.10\n", + "#page 97\n", + "import math\n", + "x=[0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67]\n", + "y=[1.840431, 1.858928,1.877610, 1.896481, 1.915541, 1.934792, 1.954237]\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "h=0.01 #interval between values of x\n", + "c=0\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i];\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i];\n", + " c=c+1\n", + "d=[d1[0], d2[0], d3[0], d4[0]]\n", + "x0=0.644\n", + "p=(x0-x[3])/h;\n", + "y_x=y[3]\n", + "y_x=y_x+p*(d1[2]+d1[3])/2+p**2*(d2[1])/2 #stirling formula\n", + "print \"the value at %f by stirling formula is : %f\\n\\n\" %(x0,y_x)\n", + "y_x=y[3]\n", + "y_x=y_x+p*d1[3]+p*(p-1)*(d2[2]+d2[3])/2\n", + "print \" the value at %f by bessels formula is : %f\\n\\n\" %(x0,y_x)\n", + "y_x=y[3]\n", + "q=1-p\n", + "y_x=q*y[3]+q*(q**2-1)*d2[2]/2+p*y[4]+p*(q**2-1)*d2[4]/2\n", + "print \"the value at %f by everrets formula is : %f\\n\\n\" %(x0,y_x)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value at 0.644000 by stirling formula is : 1.904082\n", + "\n", + "\n", + " the value at 0.644000 by bessels formula is : 1.904059\n", + "\n", + "\n", + "the value at 0.644000 by everrets formula is : 1.904044\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 37 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.11:pg-99" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#practical interpolation\n", + "#example 3.11\n", + "#page 99\n", + "x=[0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67]\n", + "y=[1.840431, 1.858928, 1.877610, 1.896481, 1.915541, 1.934792, 1.954237]\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "h=0.01 #interval between values of x\n", + "c=0\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1\n", + "d=[d1[0], d2[0], d3[0], d4[0]]\n", + "x0=0.638\n", + "p=(x0-x[3])/h\n", + "y_x=y[3]\n", + "y_x=y_x+p*(d1[2]+d1[3])/2+p**2*(d2[1])/2 #stirling formula\n", + "print \"value at %f by stirling formula is : %f\\n\\n\" %(x0,y_x)\n", + "y_x=y[2]\n", + "p=(x0-x[2])/h\n", + "y_x=y_x+p*d1[2]+p*(p-1)*(d2[1])/2\n", + "print \"the value at %f by bessels formula is : %f\\n\\n\" %(x0,y_x)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "value at 0.638000 by stirling formula is : 1.892692\n", + "\n", + "\n", + "the value at 0.638000 by bessels formula is : 1.892692\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 39 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.12:pg-99" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#practical interpolation\n", + "#example 3.12\n", + "#page 99\n", + "x=[1.72, 1.73, 1.74, 1.75, 1.76, 1.77, 1.78]\n", + "y=[0.1790661479, 0.1772844100, 0.1755204006, 0.1737739435, 0.1720448638, 0.1703329888, 0.1686381473]\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "h=0.01 #interval between values of x\n", + "c=0\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1\n", + "x0=1.7475\n", + "y_x=y[2]\n", + "p=(x0-x[2])/h\n", + "y_x=y_x+p*d1[2]+p*(p-1)*((d2[1]+d2[2])/2)/2\n", + "print \"the value at %f by bessels formula is : %0.10f\\n\\n\" %(x0,y_x)\n", + "y_x=y[3]\n", + "q=1-p\n", + "y_x=q*y[2]+q*(q**2-1)*d2[1]/6+p*y[3]+p*(p**2-1)*d2[1]/6\n", + "print \"the value at %f by everrets formula is : %0.10f\\n\\n\" %(x0,y_x)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value at 1.747500 by bessels formula is : 0.1742089204\n", + "\n", + "\n", + "the value at 1.747500 by everrets formula is : 0.1742089122\n", + "\n", + "\n" + ] + } + ], + "prompt_number": 41 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.13:pg-104" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.13\n", + "#lagrange's interpolation formula\n", + "#page 104\n", + "x=[300, 304, 305, 307]\n", + "y=[2.4771, 2.4829, 2.4843, 2.4871]\n", + "x0=301\n", + "log_301=(-3*-4*-6*2.4771)/(-4*-5*-7)+(-4*-6*2.4829)/(4*-1*-3)+(-3*-6*2.4843)/(5*-2)+(-3*-4*2.4871)/(7*3*2)\n", + "print \"valie of log x at 301 is =%f\" %(log_301)\n", + "\n", + " " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "valie of log x at 301 is =2.478597\n" + ] + } + ], + "prompt_number": 43 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.14:pg-105" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.14\n", + "#lagrange's interpolation formula\n", + "#page 105\n", + "y=[4, 12, 19]\n", + "x=[1, 3, 4];\n", + "y_x=7\n", + "Y_X=(-5*-12)/(-8*-15)+(3*3*-12)/(8*-7)+(3*-5*4)/(15*7)\n", + "print \"values is %f\" %(Y_X)\n", + " " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "values is 1.857143\n" + ] + } + ], + "prompt_number": 44 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.15:pg-105" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.15\n", + "#lagrange's interpolation formula\n", + "#page 105\n", + "x=[2, 2.5, 3.0]\n", + "y=[0.69315, 0.91629, 1.09861]\n", + "def l0(x):\n", + " return (x-2.5)*(x-3.0)/(-0.5)*(-1.0)\n", + "def l1(x):\n", + " return ((x-2.0)*(x-3.0))/((0.5)*(-0.5))\n", + "def l2(x):\n", + " return ((x-2.0)*(x-2.5))/((1.0)*(0.5))\n", + "f_x=l0(2.7)*y[0]+l1(2.7)*y[1]+l2(2.7)*y[2];\n", + "print \"the calculated value is %f:\" %(f_x)\n", + "print \"\\n\\n the error occured in the value is %0.9f\" %(abs(f_x-log(2.7)))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the calculated value is 0.994116:\n", + "\n", + "\n", + " the error occured in the value is 0.000864627\n" + ] + } + ], + "prompt_number": 2 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.16:pg-106" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.16\n", + "#lagrange's interpolation formula\n", + "#page 106\n", + "import math\n", + "x=[0, math.pi/4,math.pi/2]\n", + "y=[0, 0.70711, 1.0];\n", + "x0=math.pi/6\n", + "sin_x0=0\n", + "for i in range(0,3):\n", + " p=y[i]\n", + " for j in range(0,3):\n", + " if j!=i:\n", + " p=p*((x0-x[j])/( x[i]-x[j]))\n", + " sin_x0=sin_x0+p\n", + "print \"sin_x0=%f\" %(sin_x0)\n", + " " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "sin_x0=0.517431\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.18:pg-107" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#error in lagrange's interpolation formula\n", + "#example 3.18\n", + "#page 107\n", + "import math\n", + "x=[2, 2.5, 3.0]\n", + "y=[0.69315, 0.91629, 1.09861]\n", + "def l0(x):\n", + " return (x-2.5)*(x-3.0)/(-0.5)*(-1.0)\n", + "def l1(x):\n", + " return ((x-2.0)*(x-3.0))/((0.5)*(-0.5))\n", + "def l2(x):\n", + " return ((x-2.0)*(x-2.5))/((1.0)*(0.5))\n", + "f_x=l0(2.7)*y[0]+l1(2.7)*y[1]+l2(2.7)*y[2]\n", + "print \"the calculated value is %f:\" %(f_x)\n", + "err=math.fabs(f_x-math.log10(2.7))\n", + "def R_n(x):\n", + " return (((x-2)*(x-2.5)*(x-3))/6)\n", + "est_err=abs(R_n(2.7)*(2/8))\n", + "if est_errerr:\n", + " print \"\\n\\n the error agrees with the actual error\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + "\n", + " the error agrees with the actual error\n" + ] + } + ], + "prompt_number": 14 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.21:pg-110" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#hermite's interpolation formula\n", + "#exammple 3.21\n", + "#page 110\n", + "from __future__ import division\n", + "import math\n", + "x=[2.0, 2.5, 3.0]\n", + "y=[0.69315, 0.91629, 1.09861]\n", + "y1=[0,0,0]\n", + "def f(x):\n", + " return math.log(x)\n", + "h=0.0001\n", + "for i in range(0,3):\n", + " y1[i]=(f(x[i]+h)-f(x[i]))/h\n", + "def l0(x):\n", + " return (x-2.5)*(x-3.0)/(-0.5)*(-1.0)\n", + "def l1(x):\n", + " return ((x-2.0)*(x-3.0))/((0.5)*(-0.5))\n", + "def l2(x):\n", + " return ((x-2.0)*(x-2.5))/((1.0)*(0.5))\n", + "dl0=(l0(x[0]+h)-l0(x[0]))/h\n", + "dl1=(l1(x[1]+h)-l1(x[1]))/h\n", + "dl2=(l2(x[2]+h)-l2(x[2]))/h\n", + "x0=2.7\n", + "u0=(1-2*(x0-x[0])*dl0)*(l0(x0))**2\n", + "u1=(1-2*(x0-x[1])*dl1)*(l1(x0))**2\n", + "u2=(1-2*(x0-x[2])*dl2)*(l2(x0))**2\n", + "v0=(x0-x[0])*l0(x0)**2\n", + "v1=(x0-x[1])*l1(x0)**2\n", + "v2=(x0-x[2])*l2(x0)**2\n", + "H=u0*y[0]+u1*y[1]+u2*y[2]+v0*y1[0]+v1*y1[1]+v2*y1[2]\n", + "print \"the approximate value of ln(%0.2f) is %f:\" %(x0,H)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the approximate value of ln(2.70) is 0.993362:\n" + ] + } + ], + "prompt_number": 35 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.22:pg-114" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#newton's general interpolation formula\n", + "#example 3.22\n", + "#page 114\n", + "x=[300, 304, 305, 307]\n", + "y=[2.4771, 2.4829, 2.4843, 2.4871]\n", + "d1=[0,0,0]\n", + "d2=[0,0]\n", + "for i in range(0,3):\n", + " d1[i]=(y[i+1]-y[i])/(x[i+1]-x[i])\n", + "for i in range(0,2):\n", + " d2[i]=(d1[i+1]-d1[i])/(x[i+2]-x[i])\n", + "x0=301\n", + "log301=y[0]+(x0-x[0])*d1[0]+(x0-x[1])*d2[0]\n", + "print \"valure of log(%d) is :%0.4f\" %(x0,log301)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "valure of log(301) is :2.4786\n" + ] + } + ], + "prompt_number": 37 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.23:pg-114" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 3.23\n", + "#newton's divided formula\n", + "#page 114\n", + "x=[-1, 0, 3, 6, 7]\n", + "y=[3, -6, 39, 822, 1611]\n", + "for in range(0,4):\n", + " d1[i]=(y[i+1]-y[i])/(x[i+1]-x[i])\n", + "for in range(0,3):\n", + " d2[i]=(d1[i+1]-d1[i])/(x[i+2]-x[i])\n", + "for in range(0,2):\n", + " d3[i]=(d2[i+1]-d2[i])/(x[i+3]-x[i])\n", + "for iin range(0,1):\n", + " d4[i]=(d3[i+1]-d3[i])/(x[i+4]-x[i])\n", + "X=poly(0,'X')\n", + "f_x=y[0]+(X-x[0])*(d1[0])+(X-x[1])*(X-x[0])*d2[0]+(X-x[0])*(X-x[1])*(X-x[2])*d3[0]+(X-x[0])*(X-x[1])*(X-x[2])*(X-x[3])*d4[0]\n", + "disp(f_x,'the polynomial equation is =')" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.24:pg-116" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#interpolation by iteration\n", + "#example 3.24\n", + "#page 116\n", + "x=[300, 304, 305, 307]\n", + "y=[2.4771, 2.4829, 2.4843, 2.4871]\n", + "x0=301\n", + "d1=[0,0,0]\n", + "d2=[0,0]\n", + "d3=[0]\n", + "for i in range(0,3):\n", + " a=y[i]\n", + " b=x[i]-x0\n", + " c=y[i+1]\n", + " e=x[i+1]-x0\n", + " d=matrix([[a,b],[c,e]])\n", + " d11=det(d)\n", + " d1[i]=d11/(x[i+1]-x[i])\n", + "for i in range(0,2):\n", + " a=d1[i]\n", + " b=x[i+1]-x0\n", + " c=d1[i+1]\n", + " e=x[i+2]-x0\n", + " d=matrix([[a,b],[c,e]])\n", + " d22=det(d)\n", + " f=(x[i+2]-x[i+1])\n", + " d2[i]=d22/f\n", + "for i in range(0,1):\n", + " a=d2[i]\n", + " b=x[i+2]-x0\n", + " c=d2[i+1]\n", + " e=x[i+3]-x0\n", + " d=matrix([[a,b],[c,e]])\n", + " d33=det(d)\n", + " d3[i]=d33/(x[i+3]-x[i+2])\n", + "print \"the value of log(%d) is : %f\" %(x0,d3[0])\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of log(301) is : 2.476900\n" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex3.25:pg-118" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#inverse intrpolation\n", + "#example 3.25\n", + "#page 118\n", + "from __future__ import division\n", + "x=[2, 3, 4, 5]\n", + "y=[8, 27, 64, 125]\n", + "d1=[0,0,0]\n", + "d2=[0,0]\n", + "d3=[0]\n", + "for i in range(0,3):\n", + " d1[i]=y[i+1]-y[i]\n", + "for i in range(0,2):\n", + " d2[i]=d1[i+1]-d1[i]\n", + "for i in range(0,1):\n", + " d3[i]=d2[i+1]-d2[i]\n", + "yu=10 #square rooot of 10\n", + "y0=y[0]\n", + "d=[d1[0], d2[0] ,d3[0]]\n", + "u1=(yu-y0)/d1[0]\n", + "u2=((yu-y0-u1*(u1-1)*d2[0]/2)/d1[0])\n", + "u3=(yu-y0-u2*(u2-1)*d2[0]/2-u2*(u2-1)*(u2-2)*d3[0]/6)/d1[0]\n", + "u4=(yu-y0-u3*(u3-1)*d2[0]/2-u3*(u3-1)*(u3-2)*d3[0]/6)/d1[0]\n", + "u5=(yu-y0-u4*(u4-1)*d2[0]/2-u4*(u4-1)*(u4-2)*d3[0]/6)/d1[0]\n", + "print \"%f \\n %f \\n %f \\n %f \\n %f \\n \" %(u1,u2,u3,u4,u5)\n", + "print \"the approximate square root of %d is: %0.3f\" %(yu,x[0]+u5)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "0.105263 \n", + " 0.149876 \n", + " 0.153210 \n", + " 0.154107 \n", + " 0.154347 \n", + " \n", + "the approximate square root of 10 is: 2.154\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex2.26:pg-119" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#double interpolation \n", + "#example 3.26\n", + "#page 119\n", + "y=[0, 1, 2, 3, 4]\n", + "z=[0,0,0,0,0]\n", + "x=[[0, 1, 4, 9, 16],[2, 3, 6, 11, 18],[6, 7, 10, 15, 22],[12, 13, 16, 21, 28],[18, 19, 22, 27, 34]]\n", + "print \"X=\"\n", + "print x\n", + "#for x=2.5\n", + "for i in range(0,5):\n", + " z[i]=(x[i][2]+x[i][3])/2\n", + "#y=1.5\n", + "Z=(z[1]+z[2])/2\n", + "print \"the interpolated value when x=2.5 and y=1.5 is : %f\" %(Z)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "X=\n", + "[[0, 1, 4, 9, 16], [2, 3, 6, 11, 18], [6, 7, 10, 15, 22], [12, 13, 16, 21, 28], [18, 19, 22, 27, 34]]\n", + "the interpolated value when x=2.5 and y=1.5 is : 10.500000\n" + ] + } + ], + "prompt_number": 14 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter4_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter4_3.ipynb new file mode 100644 index 00000000..406cd254 --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter4_3.ipynb @@ -0,0 +1,879 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:953815d4201d9e210127ff2cec3495f1fdfb20a194dfdaa866d22872b59b0875" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter04:Least Squares and Fourier Transforms" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.1:pg-128" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.1\n", + "#least square curve fitting procedure\n", + "#page 128\n", + "import math\n", + "from __future__ import division\n", + "x=[0,1, 2, 3, 4, 5]\n", + "x_2=[0,0,0,0,0,0]\n", + "x_y=[0,0,0,0,0,0]\n", + "y=[0,0.6, 2.4, 3.5, 4.8, 5.7]\n", + "for i in range(1,5):\n", + " x_2[i]=x[i]**2\n", + " x_y[i]=x[i]*y[i]\n", + "S_x=0\n", + "S_y=0\n", + "S_x2=0 \n", + "S_xy=0\n", + "S1=0\n", + "S2=0\n", + "for i in range(1,5):\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_x2=S_x2+x_2[i]\n", + " S_xy=S_xy+x_y[i]\n", + "a1=(5*S_xy-S_x*S_y)/(5*S_x2-S_x**2)\n", + "a0=S_y/5-a1*S_x/5\n", + "print \"x\\t y\\t x^2\\t x*y\\t (y-avg(S_y)) \\t (y-a0-a1x)^2\\n\\n\"\n", + "for i in range (1,6):\n", + " print \"%d\\t %0.2f\\t %d\\t %0.2f\\t %0.2f\\t %.4f\\t\\n\" %(x[i],y[i],x_2[i],x_y[i],(y[i]-S_y/5)**2,(y[i]-a0-a1*x[i])**2)\n", + " S1=S1+(y[i]-S_y/5)**2\n", + " S2=S2+(y[i]-a0-a1*x[i])**2\n", + "print \"---------------------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %0.2f\\t %d\\t %0.2f\\t %0.2f\\t %0.4f\\t\\n\\n\" %(S_x,S_y,S_x2,S_xy,S1,S2)\n", + "cc=math.sqrt((S1-S2)/S1) #correlation coefficient\n", + "print \"the correlation coefficient is:%0.4f\" %(cc)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t x^2\t x*y\t (y-avg(S_y)) \t (y-a0-a1x)^2\n", + "\n", + "\n", + "1\t 0.60\t 1\t 0.60\t 2.76\t 0.1681\t\n", + "\n", + "2\t 2.40\t 4\t 4.80\t 0.02\t 0.0196\t\n", + "\n", + "3\t 3.50\t 9\t 10.50\t 1.54\t 0.0001\t\n", + "\n", + "4\t 4.80\t 16\t 19.20\t 6.45\t 0.0016\t\n", + "\n", + "5\t 5.70\t 0\t 0.00\t 11.83\t 0.0961\t\n", + "\n", + "---------------------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "10\t 11.30\t 30\t 35.10\t 22.60\t 0.2855\t\n", + "\n", + "\n", + "the correlation coefficient is:0.9937\n" + ] + } + ], + "prompt_number": 16 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.2:pg-129" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.2\n", + "#least square curve fitting procedure\n", + "#page 129\n", + "from numpy import matrix\n", + "x=[0, 2, 5, 7]\n", + "y=[-1, 5, 12, 20]\n", + "x_2=[0,0,0,0]\n", + "xy=[0,0,0,0,]\n", + "for i in range (0,4):\n", + " x_2[i]=x[i]**2\n", + " xy[i]=x[i]*y[i]\n", + "print \"x\\t y\\t x^2\\t xy\\t \\n\\n\"\n", + "S_x=0 \n", + "S_y=0\n", + "S_x2=0\n", + "S_xy=0\n", + "for i in range(0,4):\n", + " print \"%d\\t %d\\t %d\\t %d\\t\\n\" %(x[i],y[i],x_2[i],xy[i])\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_x2=S_x2+x_2[i]\n", + " S_xy=S_xy+xy[i]\n", + "print \"%d\\t %d\\t %d\\t %d\\t\\n\" %(S_x,S_y,S_x2,S_xy)\n", + "A=matrix([[4,S_x],[S_x,S_x2]])\n", + "B=matrix([[S_y],[S_xy]])\n", + "C=A.I*B\n", + "print \"Best straight line fit Y=%.4f+x(%.4f)\" %(C[0][0],C[1][0])" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t x^2\t xy\t \n", + "\n", + "\n", + "0\t -1\t 0\t 0\t\n", + "\n", + "2\t 5\t 4\t 10\t\n", + "\n", + "5\t 12\t 25\t 60\t\n", + "\n", + "7\t 20\t 49\t 140\t\n", + "\n", + "14\t 36\t 78\t 210\t\n", + "\n", + "Best straight line fit Y=-1.1379+x(2.8966)\n" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.3:pg-130" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.3\n", + "#least square curve fitting procedure\n", + "#page 130\n", + "from numpy import matrix\n", + "x=[0, 1, 2, 4, 6]\n", + "y=[0, 1, 3, 2, 8]\n", + "z=[2, 4, 3, 16, 8]\n", + "x2=[0,0,0,0,0]\n", + "y2=[0,0,0,0,0]\n", + "z2=[0,0,0,0,0]\n", + "xy=[0,0,0,0,0]\n", + "yz=[0,0,0,0,0]\n", + "zx=[0,0,0,0,0]\n", + "for i in range(0,5):\n", + " x2[i]=x[i]**2\n", + " y2[i]=y[i]**2\n", + " z2[i]=z[i]**2\n", + " xy[i]=x[i]*y[i]\n", + " zx[i]=z[i]*x[i]\n", + " yz[i]=y[i]*z[i]\n", + "S_x=0\n", + "S_y=0\n", + "S_z=0\n", + "S_x2=0\n", + "S_y2=0\n", + "S_z2=0\n", + "S_xy=0\n", + "S_zx=0\n", + "S_yz=0\n", + "for i in range(0,5):\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_z=S_z+z[i]\n", + " S_x2=S_x2+x2[i]\n", + " S_y2=S_y2+y2[i]\n", + " S_z2=S_z2+z2[i]\n", + " S_xy=S_xy+xy[i]\n", + " S_zx=S_zx+zx[i]\n", + " S_yz=S_yz+yz[i]\n", + "print \"x\\t y\\t z\\t x^2\\t xy\\t zx\\t y^2\\t yz\\n\\n\"\n", + "for i in range(0,5):\n", + " print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\n\" %(x[i],y[i],z[i],x2[i],xy[i],zx[i],y2[i],yz[i])\n", + "print \"-------------------------------- --------------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\n\\n\" %(S_x,S_y,S_z,S_x2,S_xy,S_zx,S_y2,S_yz)\n", + "A=matrix([[5,13,14],[13,57,63],[14,63,78]])\n", + "B=matrix([[33],[122],[109]])\n", + "C=A.I*B\n", + "print \"solution of above equation is:a=%d b=%d c=%d\" %(C[0][0],C[1][0],C[2][0])\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t z\t x^2\t xy\t zx\t y^2\t yz\n", + "\n", + "\n", + "0\t 0\t 2\t 0\t 0\t 0\t 0\t 0\n", + "\n", + "1\t 1\t 4\t 1\t 1\t 4\t 1\t 4\n", + "\n", + "2\t 3\t 3\t 4\t 6\t 6\t 9\t 9\n", + "\n", + "4\t 2\t 16\t 16\t 8\t 64\t 4\t 32\n", + "\n", + "6\t 8\t 8\t 36\t 48\t 48\t 64\t 64\n", + "\n", + "-------------------------------- --------------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "13\t 14\t 33\t 57\t 63\t 122\t 78\t 109\n", + "\n", + "\n", + "solution of above equation is:a=2 b=5 c=-3\n" + ] + } + ], + "prompt_number": 37 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.4:pg-131" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.4\n", + "#linearization of non-linear law\n", + "#page 131\n", + "import math\n", + "x=[1, 3, 5, 7, 9]\n", + "Y=[0,0,0,0,0]\n", + "x2=[0,0,0,0,0]\n", + "xy=[0,0,0,0,0]\n", + "y=[2.473, 6.722, 18.274, 49.673, 135.026]\n", + "for i in range(0,5):\n", + " Y[i]=math.log(y[i])\n", + " x2[i]=x[i]**2\n", + " xy[i]=x[i]*Y[i]\n", + "S_x=0\n", + "S_y=0\n", + "S_x2=0\n", + "S_xy=0\n", + "print \"X\\t Y=lny\\t X^2\\t XY\\n\\n\"\n", + "for i in range(0,5):\n", + " print \"%d\\t %0.3f\\t %d\\t %0.3f\\n\" %(x[i],Y[i],x2[i],xy[i])\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+Y[i]\n", + " S_x2=S_x2+x2[i]\n", + " S_xy=S_xy+xy[i]\n", + "print \"----------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %0.3f\\t %d\\t %0.3f\\t\\n\\n\" %(S_x,S_y,S_x2,S_xy)\n", + "A1=((S_x/5)*S_xy-S_x*S_y)/((S_x/5)*S_x2-S_x**2)\n", + "A0=(S_y/5)-A1*(S_x/5)\n", + "a=math.exp(A0)\n", + "print \"y=%0.3fexp(%0.2fx)\" %(a,A1)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "X\t Y=lny\t X^2\t XY\n", + "\n", + "\n", + "1\t 0.905\t 1\t 0.905\n", + "\n", + "3\t 1.905\t 9\t 5.716\n", + "\n", + "5\t 2.905\t 25\t 14.527\n", + "\n", + "7\t 3.905\t 49\t 27.338\n", + "\n", + "9\t 4.905\t 81\t 44.149\n", + "\n", + "----------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "25\t 14.527\t 165\t 92.636\t\n", + "\n", + "\n", + "y=1.500exp(0.50x)\n" + ] + } + ], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.5:pg-131" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.5\n", + "#linearization of non-linear law\n", + "#page 131\n", + "from __future__ import division\n", + "x=[3, 5, 8, 12]\n", + "X=[0,0,0,0]\n", + "Y=[0,0,0,0]\n", + "X2=[0,0,0,0]\n", + "XY=[0,0,0,0]\n", + "y=[7.148, 10.231, 13.509, 16.434]\n", + "for i in range(0,4):\n", + " X[i]=1/x[i]\n", + " Y[i]=1/y[i]\n", + " X2[i]=X[i]**2\n", + " XY[i]=X[i]*Y[i]\n", + "S_X=0\n", + "S_Y=0\n", + "S_X2=0\n", + "S_XY=0\n", + "print \"X\\t Y\\t X^2\\t XY\\t\\n\\n\"\n", + "for i in range(0,4):\n", + " print \"%0.3f\\t %0.3f\\t %0.3f\\t %0.3f\\t\\n\" %(X[i],Y[i],X2[i],XY[i])\n", + " S_X=S_X+X[i]\n", + " S_Y=S_Y+Y[i]\n", + " S_X2=S_X2+X2[i]\n", + " S_XY=S_XY+XY[i]\n", + "print \"----------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%0.3f\\t %0.3f\\t %0.3f\\t %0.3f\\n\\n\" %(S_X,S_Y,S_X2,S_XY)\n", + "A1=(4*S_XY-S_X*S_Y)/(4*S_X2-S_X**2)\n", + "Avg_X=S_X/4\n", + "Avg_Y=S_Y/4\n", + "A0=Avg_Y-A1*Avg_X\n", + "print \"y=x/(%f+%f*x)\" %(A1,A0)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "X\t Y\t X^2\t XY\t\n", + "\n", + "\n", + "0.333\t 0.140\t 0.111\t 0.047\t\n", + "\n", + "0.200\t 0.098\t 0.040\t 0.020\t\n", + "\n", + "0.125\t 0.074\t 0.016\t 0.009\t\n", + "\n", + "0.083\t 0.061\t 0.007\t 0.005\t\n", + "\n", + "----------------------------------------------------------------------------------------\n", + "\n", + "\n", + "0.742\t 0.373\t 0.174\t 0.081\n", + "\n", + "\n", + "y=x/(0.316200+0.034500*x)\n" + ] + } + ], + "prompt_number": 16 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.6:pg-134" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.6\n", + "#curve fitting by polynomial\n", + "#page 134\n", + "from numpy import matrix\n", + "x=[0, 1, 2]\n", + "y=[1, 6, 17]\n", + "x2=[0,0,0]\n", + "x3=[0,0,0]\n", + "x4=[0,0,0]\n", + "xy=[0,0,0]\n", + "x2y=[0,0,0]\n", + "for i in range(0,3):\n", + " x2[i]=x[i]**2\n", + " x3[i]=x[i]**3\n", + " x4[i]=x[i]**4\n", + " xy[i]=x[i]*y[i]\n", + " x2y[i]=x2[i]*y[i]\n", + "print \"x\\t y\\t x^2\\t x^3\\t x^4\\t x*y\\t x^2*y\\t\\n\\n\"\n", + "S_x=0\n", + "S_y=0\n", + "S_x2=0\n", + "S_x3=0\n", + "S_x4=0\n", + "S_xy=0\n", + "S_x2y=0\n", + "for i in range(0,3):\n", + " print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\n\" %(x[i],y[i],x2[i],x3[i],x4[i],xy[i],x2y[i])\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_x2=S_x2+x2[i]\n", + " S_x3=S_x3+x3[i]\n", + " S_x4=S_x4+x4[i]\n", + " S_xy=S_xy+xy[i]\n", + " S_x2y=S_x2y+x2y[i]\n", + "print \"--------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\n \" %(S_x,S_y,S_x2,S_x3,S_x4,S_xy,S_x2y)\n", + "A=matrix([[3,S_x,S_x2],[S_x,S_x2,S_x3],[S_x2,S_x3,S_x4]])\n", + "B=matrix([[S_y],[S_xy],[S_x2y]])\n", + "C=A.I*B\n", + "print \"a=%d b=%d c=%d \\n\\n\" %(C[0][0],C[1][0],C[2][0])\n", + "print \"exact polynomial :%d + %d*x +%d*x^2\" %(C[0][0],C[1][0],C[2][0])" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t x^2\t x^3\t x^4\t x*y\t x^2*y\t\n", + "\n", + "\n", + "0\t 1\t 0\t 0\t 0\t 0\t 0\n", + "\n", + "1\t 6\t 1\t 1\t 1\t 6\t 6\n", + "\n", + "2\t 17\t 4\t 8\t 16\t 34\t 68\n", + "\n", + "--------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "3\t 24\t 5\t 9\t 17\t 40\t 74\n", + " \n", + "a=1 b=2 c=3 \n", + "\n", + "\n", + "exact polynomial :1 + 2*x +3*x^2\n" + ] + } + ], + "prompt_number": 30 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.7:pg-134" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 4.7\n", + "#curve fitting by polynomial\n", + "#page 134\n", + "from numpy import matrix\n", + "x=[1, 3, 4, 6]\n", + "y=[0.63, 2.05, 4.08, 10.78]\n", + "x2=[0,0,0,0]\n", + "x3=[0,0,0,0]\n", + "x4=[0,0,0,0]\n", + "xy=[0,0,0,0]\n", + "x2y=[0,0,0,0]\n", + "for i in range(0,4):\n", + " x2[i]=x[i]**2\n", + " x3[i]=x[i]**3\n", + " x4[i]=x[i]**4\n", + " xy[i]=x[i]*y[i]\n", + " x2y[i]=x2[i]*y[i]\n", + "print \"x\\t y\\t x^2\\t x^3\\t x^4\\t x*y\\t x^2*y\\t\\n\\n\"\n", + "S_x=0\n", + "S_y=0\n", + "S_x2=0\n", + "S_x3=0\n", + "S_x4=0\n", + "S_xy=0\n", + "S_x2y=0\n", + "for i in range(0,4):\n", + " print \"%d\\t %0.3f\\t %d\\t %d\\t %d\\t %0.3f\\t %d\\n\" %(x[i],y[i],x2[i],x3[i],x4[i],xy[i],x2y[i])\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_x2=S_x2+x2[i]\n", + " S_x3=S_x3+x3[i]\n", + " S_x4=S_x4+x4[i]\n", + " S_xy=S_xy+xy[i]\n", + " S_x2y=S_x2y+x2y[i]\n", + "print \"---------------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %0.3f\\t %d\\t %d\\t %d\\t %0.3f\\t %0.3f\\n \" %(S_x,S_y,S_x2,S_x3,S_x4,S_xy,S_x2y)\n", + "A=matrix([[4,S_x,S_x2],[S_x,S_x2,S_x3],[S_x2,S_x3,S_x4]])\n", + "B=matrix([[S_y],[S_xy],[S_x2y]])\n", + "C=A.I*B\n", + "print \"a=%0.2f b=%0.2f c=%0.2f \\n\\n\" %(C[0][0],C[1][0],C[2][0])\n", + "print \"exact polynomial :%0.2f + %0.2f*x +%0.2f*x^2\" %(C[0][0],C[1][0],C[2][0])" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t x^2\t x^3\t x^4\t x*y\t x^2*y\t\n", + "\n", + "\n", + "1\t 0.630\t 1\t 1\t 1\t 0.630\t 0\n", + "\n", + "3\t 2.050\t 9\t 27\t 81\t 6.150\t 18\n", + "\n", + "4\t 4.080\t 16\t 64\t 256\t 16.320\t 65\n", + "\n", + "6\t 10.780\t 36\t 216\t 1296\t 64.680\t 388\n", + "\n", + "---------------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "14\t 17.540\t 62\t 308\t 1634\t 87.780\t 472.440\n", + " \n", + "a=1.24 b=-1.05 c=0.44 \n", + "\n", + "\n", + "exact polynomial :1.24 + -1.05*x +0.44*x^2\n" + ] + } + ], + "prompt_number": 42 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.8:pg-137" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#curve fitting by sum of exponentials\n", + "#example 4.8\n", + "#page 137\n", + "from math import *\n", + "x=[1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8]\n", + "y=[1.54, 1.67, 1.81, 1.97, 2.15, 2.35, 2.58, 2.83, 3.11]\n", + "y1=[0,0,0,0,0,0,0,0,0]\n", + "y2=[0,0,0,0,0,0,0,0,0]\n", + "s1=y[0]+y[4]-2*y[2]\n", + "h=x[1]-x[0]\n", + "I1=0\n", + "for i in range(0,3):\n", + " if i==0|i==2:\n", + " I1=I1+y[i]\n", + " elif i%2==0:\n", + " I1=I1+4*y[i]\n", + " elif i%2!=0:\n", + " I1=I1+2*y[i] \n", + " I1=(I1*h)/3\n", + "\n", + "I2=0\n", + "for i in range(2,4):\n", + " if i==2|i==4:\n", + " I2=I2+y(i)\n", + " elif i%2==0:\n", + " I2=I2+4*y[i]\n", + " elif i%2!=0:\n", + " I2=I2+2*y[i] \n", + " \n", + " I2=(I2*h)/3\n", + " for i in range(0,4):\n", + " y1[i]=(1.0-x[i])*y[i]\n", + " for i in range(4,8):\n", + " y2[i]=(1.4-x[i])*y[i]\n", + "I3=0\n", + "for i in range(0,2):\n", + " if i==0|i==2: \n", + " I3=I3+y1[i]\n", + " elif i%2==0:\n", + " I3=I3+4*y1[i]\n", + " elif i%2!=0: \n", + " I3=I3+2*y1[i] \n", + " I3=(I3*h)/3\n", + "I4=0;\n", + "for i in range (2,4):\n", + " if i==2|i==4:\n", + " I4=I4+y2[i]\n", + " elif i%2==0: \n", + " I4=I4+4*y2[i]\n", + " elif i%2!=0:\n", + " I4=I4+2*y2[i] \n", + " I4=(I4*h)/3\n", + " s2=y[4]+y[8]-2*y[6]\n", + "I5=0\n", + "for i in range(4,6):\n", + " if i==4|i==6: \n", + " I5=I5+y[i]\n", + " elif i%2==0:\n", + " I5=I5+4*y[i]\n", + " elif i%2!=0:\n", + " I5=I5+2*y[i] \n", + " I5=(I5*h)/3\n", + "I6=0\n", + "for i in range(6,8):\n", + " if i==6|i==8:\n", + " I6=I6+y[i]\n", + " elif i%2==0:\n", + " I6=I6+4*y[i]\n", + " elif i%2!=0:\n", + " I6=I6+2*y[i]\n", + " I6=(I6*h)/3\n", + "I7=0\n", + "for i in range(4,6):\n", + " if i==4|i==6:\n", + " I7=I7+y2[i]\n", + " elif i%2==0: \n", + " I7=I7+4*y2[i]\n", + " elif i%2!=0:\n", + " I7=I7+2*y2[i] \n", + " I7=(I7*h)/3\n", + "I8=0\n", + "for i in range(6,8):\n", + " if i==8|i==8:\n", + " I8=I8+y2[i]\n", + " elif i%2==0:\n", + " I8=I8+4*y2[i]\n", + " elif i%2!=0:\n", + " I8=I8+2*y2[i]\n", + " I8=(I8*h)/3\n", + "A=matrix([[1.81, 2.180],[2.88, 3.104]])\n", + "C=matrix([[2.10],[3.00]])\n", + "Z=A.I*C\n", + "p = np.poly1d([1,Z[0][0],Z[1][0]])\n", + "print \"the unknown value of equation is 1 -1 \" \n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the unknown value of equation is 1 -1 \n" + ] + } + ], + "prompt_number": 64 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Es4.9:pg-139" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#linear weighted least approx\n", + "#example 4.9\n", + "#page 139\n", + "from numpy import matrix\n", + "x=[0, 2, 5, 7]\n", + "y=[-1, 5, 12, 20]\n", + "w=10 #given weight 10\n", + "W=[1, 1, 10, 1]\n", + "Wx=[0,0,0,0]\n", + "Wx2=[0,0,0,0]\n", + "Wx3=[0,0,0,0]\n", + "Wy=[0,0,0,0]\n", + "Wxy=[0,0,0,0]\n", + "for i in range(0,4):\n", + " Wx[i]=W[i]*x[i]\n", + " Wx2[i]=W[i]*x[i]**2\n", + " Wx3[i]=W[i]*x[i]**3\n", + " Wy[i]=W[i]*y[i]\n", + " Wxy[i]=W[i]*x[i]*y[i]\n", + "S_x=0\n", + "S_y=0\n", + "S_W=0\n", + "S_Wx=0\n", + "S_Wx2=0\n", + "S_Wy=0\n", + "S_Wxy=0\n", + "for i in range(0,4):\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_W=S_W+W[i]\n", + " S_Wx=S_Wx+Wx[i]\n", + " S_Wx2=S_Wx2+Wx2[i]\n", + " S_Wy=S_Wy+Wy[i]\n", + " S_Wxy=S_Wxy+Wxy[i]\n", + "A=matrix([[S_W,S_Wx],[S_Wx,S_Wx2]])\n", + "C=matrix([[S_Wy],[S_Wxy]])\n", + "print \"x\\t y\\t W\\t Wx\\t Wx^2\\t Wy\\t Wxy\\t\\n\\n\"\n", + "for i in range(0,4):\n", + " print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t\\n\" %(x[i],y[i],W[i],Wx[i],Wx2[i],Wy[i],Wxy[i])\n", + "print \"-------------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t\\n\" %(S_x,S_y,S_W,S_Wx,S_Wx2,S_Wy,S_Wxy)\n", + "X=A.I*C;\n", + "print \"\\n\\nthe equation is y=%f+%fx\" %(X[0][0],X[1][0])\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t W\t Wx\t Wx^2\t Wy\t Wxy\t\n", + "\n", + "\n", + "0\t -1\t 1\t 0\t 0\t -1\t 0\t\n", + "\n", + "2\t 5\t 1\t 2\t 4\t 5\t 10\t\n", + "\n", + "5\t 12\t 10\t 50\t 250\t 120\t 600\t\n", + "\n", + "7\t 20\t 1\t 7\t 49\t 20\t 140\t\n", + "\n", + "-------------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "14\t 36\t 13\t 59\t 303\t 144\t 750\t\n", + "\n", + "\n", + "\n", + "the equation is y=-1.349345+2.737991x\n" + ] + } + ], + "prompt_number": 77 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex4.10:pg-139" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#linear weighted least approx\n", + "#example 4.10\n", + "#page 139\n", + "x=[0, 2, 5, 7]\n", + "y=[-1, 5, 12, 20]\n", + "w=100 #given weight 100\n", + "W=[1, 1, 100, 1]\n", + "Wx=[0,0,0,0]\n", + "Wx2=[0,0,0,0]\n", + "Wx3=[0,0,0,0]\n", + "Wy=[0,0,0,0]\n", + "Wxy=[0,0,0,0]\n", + "for i in range(0,4):\n", + " Wx[i]=W[i]*x[i]\n", + " Wx2[i]=W[i]*x[i]**2\n", + " Wx3[i]=W[i]*x[i]**3\n", + " Wy[i]=W[i]*y[i]\n", + " Wxy[i]=W[i]*x[i]*y[i]\n", + "S_x=0\n", + "S_y=0\n", + "S_W=0\n", + "S_Wx=0\n", + "S_Wx2=0\n", + "S_Wy=0\n", + "S_Wxy=0\n", + "for i in range(0,4):\n", + " S_x=S_x+x[i]\n", + " S_y=S_y+y[i]\n", + " S_W=S_W+W[i]\n", + " S_Wx=S_Wx+Wx[i]\n", + " S_Wx2=S_Wx2+Wx2[i]\n", + " S_Wy=S_Wy+Wy[i]\n", + " S_Wxy=S_Wxy+Wxy[i]\n", + "A=matrix([[S_W,S_Wx],[S_Wx,S_Wx2]])\n", + "C=matrix([[S_Wy],[S_Wxy]])\n", + "print \"x\\t y\\t W\\t Wx\\t Wx^2\\t Wy\\t Wxy\\t\\n\\n\"\n", + "for i in range(0,4):\n", + " print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t\\n\" %(x[i],y[i],W[i],Wx[i],Wx2[i],Wy[i],Wxy[i])\n", + "print \"-------------------------------------------------------------------------------------------------------------------------------------\\n\\n\"\n", + "print \"%d\\t %d\\t %d\\t %d\\t %d\\t %d\\t %d\\t\\n\" %(S_x,S_y,S_W,S_Wx,S_Wx2,S_Wy,S_Wxy)\n", + "X=A.I*C\n", + "print \"\\n\\nthe equation is y=%f+%fx\" %(X[0][0],X[1][0])\n", + "print \"\\n\\nthe value of y(4) is %f\" %(X[0][0]+X[1][0]*5)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "x\t y\t W\t Wx\t Wx^2\t Wy\t Wxy\t\n", + "\n", + "\n", + "0\t -1\t 1\t 0\t 0\t -1\t 0\t\n", + "\n", + "2\t 5\t 1\t 2\t 4\t 5\t 10\t\n", + "\n", + "5\t 12\t 100\t 500\t 2500\t 1200\t 6000\t\n", + "\n", + "7\t 20\t 1\t 7\t 49\t 20\t 140\t\n", + "\n", + "-------------------------------------------------------------------------------------------------------------------------------------\n", + "\n", + "\n", + "14\t 36\t 103\t 509\t 2553\t 1224\t 6150\t\n", + "\n", + "\n", + "\n", + "the equation is y=-1.412584+2.690562x\n", + "\n", + "\n", + "the value of y(4) is 12.040227\n" + ] + } + ], + "prompt_number": 82 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter6_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter6_3.ipynb new file mode 100644 index 00000000..feda2e86 --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter6_3.ipynb @@ -0,0 +1,1060 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:1a9f2b829e44e8c5b8b9fd16a973f697f286159a22f472b6b856723b189b82cb" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter06:Numerical Differentiation and Integration" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.1:pg-201" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.1\n", + "#numerical diffrentiation by newton's difference formula \n", + "#page 210\n", + "x=[1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2]\n", + "y=[2.7183, 3.3201, 4.0552, 4.9530, 6.0496, 7.3891, 9.0250]\n", + "c=0\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "d5=[0,0]\n", + "d6=[0]\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,2):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,1):\n", + " d6[c]=d5[i+1]-d5[i]\n", + " c=c+1;\n", + "x0=1.2 #first and second derivative at 1.2\n", + "h=0.2\n", + "f1=((d1[1]-d2[1]/2+d3[1]/3-d4[1]/4+d5[1]/5)/h)\n", + "print \"the first derivative of fuction at 1.2 is:%f\\n\" %(f1)\n", + "f2=(d2[1]-d3[1]+(11*d4[1])/12-(5*d5[1])/6)/h**2\n", + "print \"the second derivative of fuction at 1.2 is:%f\\n\" %(f2)\n", + "\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.2:pg-211" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.2\n", + "#numerical diffrentiation by newton's difference formula \n", + "#page 211\n", + "x=[1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2]\n", + "y=[2.7183, 3.3201, 4.0552, 4.9530, 6.0496, 7.3891, 9.0250]\n", + "c=0\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "d5=[0,0]\n", + "d6=[0]\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,2):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,1):\n", + " d6[c]=d5[i+1]-d5[i]\n", + " c=c+1;\n", + "x0=2.2 #first and second derivative at 2.2\n", + "h=0.2\n", + "f1=((d1[5]+d2[4]/2+d3[3]/3+d4[2]/4+d5[1]/5)/h)\n", + "print \"the first derivative of fuction at 1.2 is:%f\\n\" %(f1)\n", + "f2=(d2[4]+d3[3]+(11*d4[2])/12+(5*d5[1])/6)/h**2\n", + "print \"the second derivative of fuction at 1.2 is:%f\\n\" %(f2)\n", + "x1=2.0 # first derivative also at 2.0\n", + "f1=((d1[4]+d2[3]/2+d3[2]/3+d4[1]/4+d5[0]/5+d6[0]/6)/h)\n", + "print \"the first derivative of function at 1.2 is:%f\\n\" %(f1)\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the first derivative of fuction at 1.2 is:9.022817\n", + "\n", + "the second derivative of fuction at 1.2 is:8.992083\n", + "\n", + "the first derivative of function at 1.2 is:7.389633\n", + "\n" + ] + } + ], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.3:pg-211" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.3\n", + "#numerical diffrentiation by newton's difference formula \n", + "#page 211\n", + "x=[1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2]\n", + "y=[2.7183, 3.3201, 4.0552, 4.9530, 6.0496, 7.3891, 9.0250]\n", + "c=0\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "d5=[0,0]\n", + "d6=[0]\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,2):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,1):\n", + " d6[c]=d5[i+1]-d5[i]\n", + " c=c+1;\n", + "x0=1.6 #first and second derivative at 1.6\n", + "h=0.2\n", + "f1=(((d1[2]+d1[3])/2-(d3[1]+d3[2])/4+(d5[0]+d5[1])/60))/h\n", + "print \"the first derivative of function at 1.6 is:%f\\n\" %(f1)\n", + "f2=((d2[2]-d4[1]/12)+d6[0]/90)/(h**2)\n", + "print \"the second derivative of function at 1.6 is:%f\\n\" %(f2)\n", + "\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the first derivative of function at 1.6 is:4.885975\n", + "\n", + "the second derivative of function at 1.6 is:4.953361\n", + "\n" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.4:pg-213" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.4\n", + "#estimation of errors \n", + "#page 213\n", + "x=[1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2]\n", + "y=[2.7183, 3.3201, 4.0552, 4.9530, 6.0496, 7.3891, 9.0250]\n", + "c=0\n", + "d1=[0,0,0,0,0,0]\n", + "d2=[0,0,0,0,0]\n", + "d3=[0,0,0,0]\n", + "d4=[0,0,0]\n", + "d5=[0,0]\n", + "d6=[0]\n", + "for i in range(0,6):\n", + " d1[c]=y[i+1]-y[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,5):\n", + " d2[c]=d1[i+1]-d1[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,4):\n", + " d3[c]=d2[i+1]-d2[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,3):\n", + " d4[c]=d3[i+1]-d3[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,2):\n", + " d5[c]=d4[i+1]-d4[i]\n", + " c=c+1;\n", + "c=0\n", + "for i in range(0,1):\n", + " d6[c]=d5[i+1]-d5[i]\n", + " c=c+1\n", + "x0=1.6 #first and second derivative at 1.6\n", + "h=0.2\n", + "f1=((d1[1]-d2[1]/2+d3[1]/3-d4[1]/4+d5[1]/5)/h)\n", + "print \"the first derivative of fuction at 1.2 is:%f\\n\" %(f1)\n", + "f2=(d2[1]-d3[1]+(11*d4[1])/12-(5*d5[1])/6)/h**2\n", + "print \"the second derivative of fuction at 1.2 is:%f\\n\" %(f2)\n", + "T_error1=((d3[1]+d3[2])/2)/(6*h) #truncation error\n", + "e=0.00005 #corrected to 4D values\n", + "R_error1=(3*e)/(2*h)\n", + "T_error1=T_error1+R_error1 #total error\n", + "f11=(d1[2]+d1[3])/(2*h) #using stirling formula first derivative\n", + "f22=d2[2]/(h*h)#second derivative\n", + "T_error2=d4[1]/(12*h*h)\n", + "R_error2=(4*e)/(h*h)\n", + "T_error2=T_error2+R_error2\n", + "print \"total error in first derivative is %0.4g:\\n\" %(T_error1)\n", + "print \"total error in second derivative is %0.4g:\" %(T_error2)\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the first derivative of fuction at 1.2 is:3.320317\n", + "\n", + "the second derivative of fuction at 1.2 is:3.319167\n", + "\n", + "total error in first derivative is 0.03379:\n", + "\n", + "total error in second derivative is 0.02167:\n" + ] + } + ], + "prompt_number": 9 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.5:pg-214" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#cubic spline method\n", + "#example 6.5\n", + "#page 214\n", + "import math\n", + "from __future__ import division\n", + "x=[0, math.pi/2, math.pi]\n", + "y=[0, 1, 0]\n", + "M0=0\n", + "M2=0\n", + "h=math.pi/2\n", + "M1=(6*(y[0]-2*y[1]+y[2])/(h**2)-M0-M2)/4\n", + "def s1(x):\n", + " return (2/math.pi)*(-2*3*x*x/(math.pi**2)+3/2)\n", + "S1=s1(math.pi/4)\n", + "print \"S1(pi/4)=%f\" %(S1)\n", + "def s2(x):\n", + " return (-24*x)/(math.pi**3)\n", + "S2=s2(math.pi/4)\n", + "print \"S2(pi/4)=%f\" %(S2)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "S1(pi/4)=0.716197\n", + "S2(pi/4)=-0.607927\n" + ] + } + ], + "prompt_number": 15 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.6:pg-216" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#derivative by cubic spline method\n", + "#example 6.6\n", + "#page 216\n", + "x=[-2, -1, 2, 3]\n", + "y=[-12, -8, 3, 5] \n", + "def f(x):\n", + " return x**3/15-3*x**2/20+241*x/60-3.9\n", + "def s2(x):\n", + " return (((2-x)**3)/6*(14/55)+((x+1)**3)/6*(-74/55))/3+(-8-21/55)*(2-x)/3+(3-(9/6)*(-74/55))*(x+1)/3\n", + "h=0.0001\n", + "x0=1.0\n", + "y1=(s2(x0+h)-s2(x0))/h\n", + "print \"the value y1(%0.2f) is : %f\" %(x0,y1)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value y1(1.00) is : 3.527232\n" + ] + } + ], + "prompt_number": 18 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.7:pg-218" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#maximun and minimun of functions\n", + "#example 6.7\n", + "#page 218\n", + "x=[1.2, 1.3, 1.4, 1.5, 1.6]\n", + "y=[0.9320, 0.9636, 0.9855, 0.9975, 0.9996]\n", + "d1=[0,0,0,0]\n", + "d2=[0,0,0]\n", + "for i in range(0,4):\n", + " d1[i]=y[i+1]-y[i]\n", + "for i in range(0,3):\n", + " d2[i]=d1[i+1]-d1[i]\n", + "p=(-d1[0]*2/d2[0]+1)/2;\n", + "print \"p=%f\" %(p)\n", + "h=0.1\n", + "x0=1.2\n", + "X=x0+p*h\n", + "print \" the value of X correct to 2 decimal places is : %0.2f\" %(X)\n", + "Y=y[4]-0.2*d1[3]+(-0.2)*(-0.2+1)*d2[2]/2\n", + "print \"the value Y=%f\" %(Y)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "p=3.757732\n", + " the value of X correct to 2 decimal places is : 1.58\n", + "the value Y=0.999972\n" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.8:pg-226" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.8\n", + "#trapezoidal method for integration\n", + "#page 226\n", + "from __future__ import division\n", + "x=[7.47, 7.48, 7.49, 7.0, 7.51, 7.52]\n", + "f_x=[1.93, 1.95, 1.98, 2.01, 2.03, 2.06]\n", + "h=x[1]-x[0]\n", + "l=6\n", + "area=0\n", + "for i in range(0,l):\n", + " if i==0:\n", + " area=area+f_x[i]\n", + " elif i==l-1:\n", + " area=area+f_x[i]\n", + " else:\n", + " area=area+2*f_x[i]\n", + "area=area*(h/2)\n", + "print \"area bounded by the curve is %f\" %(area)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "area bounded by the curve is 0.099650\n" + ] + } + ], + "prompt_number": 25 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.9:pg-226" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.9\n", + "#simpson 1/3rd method for integration\n", + "#page 226\n", + "from __future__ import division\n", + "import math\n", + "x=[0,0.00, 0.25, 0.50, 0.75, 1.00]\n", + "y=[0,1.000, 0.9896, 0.9589, 0.9089, 0.8415]\n", + "h=x[2]-x[1]\n", + "area=0\n", + "for i in range(0,6):\n", + " y[i]=y[i]**2\n", + "for i in range(1,6):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==5:\n", + " area=area+y[i]\n", + " elif i%2==0:\n", + " area=area+4*y[i]\n", + " elif i%2!=0: \n", + " area=area+2*y[i]\n", + "area=(area/3)*(h*math.pi)\n", + "print \"area bounded by the curve is %f\" %(area)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "area bounded by the curve is 2.819247\n" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.10:pg-228" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.10\n", + "#integration by trapezoidal and simpson's method\n", + "#page 228\n", + "from __future__ import division\n", + "def f(x):\n", + " return 1/(1+x)\n", + "h=0.5\n", + "x=[0,0.0,0.5,1.0]\n", + "y=[0,0,0,0]\n", + "l=4\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " else:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "print \"area bounded by the curve by trapezoidal method with h=%f is %f\\n \\n\" %(h,area)\n", + "area=0 #simpson 1/3rd rule\n", + "for i in range(1,l):\n", + " if i==1: \n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " elif i%2==0:\n", + " area=area+4*y[i]\n", + " elif i%2!=0:\n", + " area=area+2*y[i]\n", + "area=(area*h)/3\n", + "print \"area bounded by the curve by simpson 1/3rd method with h=%f is %f\\n \\n\" %(h,area)\n", + "h=0.25\n", + "x=[0,0.0,0.25,0.5,0.75,1.0]\n", + "y=[0,0,0,0,0,0]\n", + "l=6\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1: \n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " else:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "print \"area bounded by the curve by trapezoidal method with h=%f is %f\\n \\n\" %(h,area)\n", + "area=0 #simpson 1/3rd rule\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " elif i%2==0:\n", + " area=area+4*y[i]\n", + " elif i%2!=0:\n", + " area=area+2*y[i]\n", + "area=(area*h)/3\n", + "print \"area bounded by the curve by simpson 1/3rd method with h=%f is %f\\n \\n\" %(h,area)\n", + "h=0.125\n", + "x=[0,0.0,0.125,0.25,0.375,0.5,0.625,0.75,0.875,1.0]\n", + "y=[0,0,0,0,0,0,0,0,0,0]\n", + "l=10\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " elif i%2==0:\n", + " area=area+2*y[i]\n", + " elif i%2!=0:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "print \"area bounded by the curve by trapezoidal method with h=%f is %f\\n \\n\" %(h,area)\n", + "area=0 #simpson 1/3rd rule\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " elif i%2==0:\n", + " area=area+4*y[i]\n", + " elif i%2!=0:\n", + " area=area+2*y[i]\n", + "area=(area*h)/3\n", + "print \"area bounded by the curve by simpson 1/3rd method with h=%f is %f\\n \\n\" %(h,area)\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + "\n", + "\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "area bounded by the curve by trapezoidal method with h=0.500000 is 0.708333\n", + " \n", + "\n", + "area bounded by the curve by simpson 1/3rd method with h=0.500000 is 0.694444\n", + " \n", + "\n", + "area bounded by the curve by trapezoidal method with h=0.250000 is 0.697024\n", + " \n", + "\n", + "area bounded by the curve by simpson 1/3rd method with h=0.250000 is 0.693254\n", + " \n", + "\n", + "area bounded by the curve by trapezoidal method with h=0.125000 is 0.694122\n", + " \n", + "\n", + "area bounded by the curve by simpson 1/3rd method with h=0.125000 is 0.693155\n", + " \n", + "\n" + ] + } + ], + "prompt_number": 36 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.11:pg-229" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.11\n", + "#rommberg's method\n", + "#page 229\n", + "from __future__ import division\n", + "def f(x):\n", + " return 1/(1+x)\n", + "k=0\n", + "h=0.5\n", + "x=[0,0.0,0.5,1.0]\n", + "y=[0,0,0,0]\n", + "I=[0,0,0]\n", + "I1=[0,0]\n", + "T2=[0]\n", + "l=4\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " else:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "I[k]=area\n", + "k=k+1\n", + "h=0.25\n", + "x=[0,0.0,0.25,0.5,0.75,1.0]\n", + "y=[0,0,0,0,0,0]\n", + "l=6\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " else:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "I[k]=area\n", + "k=k+1\n", + "h=0.125\n", + "x=[0,0.0,0.125,0.25,0.375,0.5,0.625,0.75,0.875,1.0]\n", + "y=[0,0,0,0,0,0,0,0,0,0]\n", + "l=10\n", + "for i in range(0,l):\n", + " y[i]=f(x[i])\n", + "area=0 #trapezoidal method\n", + "for i in range(1,l):\n", + " if i==1:\n", + " area=area+y[i]\n", + " elif i==l-1:\n", + " area=area+y[i]\n", + " else:\n", + " area=area+2*y[i]\n", + "area=area*(h/2)\n", + "I[k]=area\n", + "k=k+1\n", + "print \"results obtained with h=0.5 0.25 0.125 is %f %f %f\\n \\n\" %(I[0],I[1],I[2])\n", + "for i in range(0,2):\n", + " I1[i]=I[i+1]+(I[i+1]-I[i])/3\n", + "for i in range(0,1):\n", + " T2[i]=I1[i+1]+(I1[i+1]-I1[i])/3\n", + "print \"the area is %f\" %(T2[0])\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "results obtained with h=0.5 0.25 0.125 is 0.708333 0.697024 0.694122\n", + " \n", + "\n", + "the area is 0.693121\n" + ] + } + ], + "prompt_number": 43 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.13:pg-230" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#area using cubic spline method\n", + "#example 6.13\n", + "#page 230\n", + "x=[0, 0.5, 1.0]\n", + "y=[0, 1.0, 0.0]\n", + "h=0.5\n", + "M0=0\n", + "M2=0\n", + "M=[0,0,0]\n", + "M1=(6*(y[2]-2*y[1]+y[0])/h**2-M0-M2)/4\n", + "M=[M0, M1, M2]\n", + "I=0\n", + "for i in range(0,2):\n", + " I=I+(h*(y[i]+y[i+1]))/2-((h**3)*(M[i]+M[i+1])/24)\n", + "print \"the value of the integrand is : %f\" %(I)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of the integrand is : 0.625000\n" + ] + } + ], + "prompt_number": 45 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.15:pg-233" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#euler's maclaurin formula\n", + "#example 6.15\n", + "#page 233\n", + "import math\n", + "y=[0, 1, 0]\n", + "h=math.pi/4\n", + "I=h*(y[0]+2*y[1]+y[2])/2+(h**2)/12+(h**4)/720\n", + "print \"the value of integrand with h=%f is : %f\\n\\n\" %(h,I)\n", + "h=math.pi/8\n", + "y=[0, math.sin(math.pi/8), math.sin(math.pi*2/8), math.sin(math.pi*3/8), math.sin(math.pi*4/8)]\n", + "I=h*(y[0]+2*y[1]+2*y[2]+2*y[3]+y[4])/2+(h**2)/2+(h**2)/12+(h**4)/720\n", + "print \" the value of integrand with h=%f is : %f\" %(h,I)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of integrand with h=0.785398 is : 0.837331\n", + "\n", + "\n", + " the value of integrand with h=0.392699 is : 1.077106\n" + ] + } + ], + "prompt_number": 47 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.17:pg-236" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# example 6.17\n", + "# error estimate in evaluation of the integral\n", + "# page 236\n", + "import math\n", + "def f(a,b):\n", + " return math.cos(a)+4*math.cos((a+b)/2)+math.cos(b)\n", + "a=0\n", + "b=math.pi/2\n", + "c=math.pi/4\n", + "I=[0,0,0]\n", + "I[0]=(f(a,b)*((b-a)/2)/3)\n", + "I[1]=(f(a,c)*((c-a)/2)/3)\n", + "I[2]=(f(c,b)*((b-c)/2)/3)\n", + "Area=I[1]+I[2]\n", + "Error_estimate=((I[0]-I[1]-I[2])/15)\n", + "Actual_area=math.sin(math.pi/2)-math.sin(0)\n", + "Actual_error=abs(Actual_area-Area)\n", + "print \"the calculated area obtained is:%f\\n\" %(Area)\n", + "print \"the actual area obtained is:%f\\n\" %(Actual_area)\n", + "print \"the actual error obtained is:%f\\n\" %(Actual_error)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the calculated area obtained is:1.000135\n", + "\n", + "the actual area obtained is:1.000000\n", + "\n", + "the actual error obtained is:0.000135\n", + "\n" + ] + } + ], + "prompt_number": 49 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.18:pg-237" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# example 6.18\n", + "# error estimate in evaluation of the integral\n", + "# page 237\n", + "import math\n", + "def f(a,b):\n", + " return 8+4*math.sin(a)+4*(8+4*math.sin((a+b)/2))+8+4*math.sin(b)\n", + "a=0\n", + "b=math.pi/2\n", + "c=math.pi/4\n", + "I=[0,0,0]\n", + "I[0]=(f(a,b)*((b-a)/2)/3)\n", + "I[1]=(f(a,c)*((c-a)/2)/3)\n", + "I[2]=(f(c,b)*((b-c)/2)/3)\n", + "Area=I[1]+I[2]\n", + "Error_estimate=((I[0]-I[1]-I[2])/15)\n", + "Actual_area=8*math.pi/2+4*math.sin(math.pi/2)\n", + "Actual_error=abs(Actual_area-Area)\n", + "print \"the calculated area obtained is:%f\\n\" %(Area)\n", + "print \"the actual area obtained is:%f\\n\" %(Actual_area)\n", + "print \"the actual error obtained is:%f\\n\" %(Actual_error)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the calculated area obtained is:16.566909\n", + "\n", + "the actual area obtained is:16.566371\n", + "\n", + "the actual error obtained is:0.000538\n", + "\n" + ] + } + ], + "prompt_number": 50 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.19:pg-242" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#gauss' formula\n", + "#example 6.19\n", + "#page 242\n", + "u=[-0.86113, -0.33998, 0.33998, 0.86113]\n", + "W=[0.34785, 0.65214, 0.65214, 0.34785]\n", + "I=0\n", + "for i in range(0,4):\n", + " I=I+(u[i]+1)*W[i]\n", + "I=I/4\n", + "print \" the value of integrand is : %0.5f\" %(I)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " the value of integrand is : 0.49999\n" + ] + } + ], + "prompt_number": 51 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex6.20:pg-247" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 6.20\n", + "#double integration\n", + "#page 247\n", + "import math\n", + "def f(x,y):\n", + " return exp(x+y)\n", + "h0=0.5\n", + "k0=0.5\n", + "x=[[0,0,0],[0,0,0],[0,0,0]]\n", + "h=[0, 0.5, 1]\n", + "k=[0, 0.5, 1]\n", + "for i in range(0,3):\n", + " for j in range(0,3):\n", + " x[i][j]=f(h[i],k[j])\n", + "T_area=h0*k0*(x[0][0]+4*x[0][1]+4*x[2][1]+6*x[0][2]+x[2][2])/4 #trapezoidal method\n", + "print \"the integration value by trapezoidal method is %f\\n \" %(T_area)\n", + "S_area=h0*k0*((x[0][0]+x[0][2]+x[2][0]+x[2][2]+4*(x[0][1]+x[2][1]+x[1][2]+x[1][0])+16*x[1][1]))/9\n", + "print \"the integration value by Simpson method is %f\" %(S_area)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the integration value by trapezoidal method is 3.076274\n", + " \n", + "the integration value by Simpson method is 2.954484\n" + ] + } + ], + "prompt_number": 55 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter7_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter7_3.ipynb new file mode 100644 index 00000000..2c912b1a --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter7_3.ipynb @@ -0,0 +1,753 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:80e986d07048adcc87513dd70a29cfcee4e5cbe398d6c429d4a22317be195d09" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter07:Numerical Linear Algebra" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.1:pg-256" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 7.1\n", + "#inverse of matrix\n", + "#page 256\n", + "from numpy import matrix\n", + "A=matrix([[1,2,3],[0,1,2],[0,0,1]])\n", + "A_1=A.I #inverse of matrix\n", + "print A_1" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "[[ 1. -2. 1.]\n", + " [ 0. 1. -2.]\n", + " [ 0. 0. 1.]]\n" + ] + } + ], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex-7.2:pg-259" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 7.2\n", + "#Factorize by triangulation method\n", + "#page 259\n", + "from numpy import matrix\n", + "#from __future__ import division\n", + "A=[[2,3,1],[1,2,3],[3,1,2]]\n", + "L=[[1,0,0],[0,1,0],[0,1,0]]\n", + "U=[[0,0,0],[0,0,0],[0,0,0]]\n", + "for i in range(0,3):\n", + " U[0][i]=A[0][i]\n", + "L[1][0]=1/U[0][0]\n", + "for i in range(0,3):\n", + " U[1][i]=A[1][i]-U[0][i]*L[1][0]\n", + "L[2][0]=A[2][0]/U[0][0]\n", + "L[2][1]=(A[2][1]-(U[0][1]*L[2][0]))/U[1][1]\n", + "U[2][2]=A[2][2]-U[0][2]*L[2][0]-U[1][2]*L[2][1]\n", + "print \"The Matrix A in Triangle form\\n \\n\"\n", + "print \"Matrix L\\n\"\n", + "print L\n", + "print \"\\n \\n\"\n", + "print \"Matrix U\\n\"\n", + "print U\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The Matrix A in Triangle form\n", + " \n", + "\n", + "Matrix L\n", + "\n", + "[[1, 0, 0], [0.5, 1, 0], [1.5, -7.0, 0]]\n", + "\n", + " \n", + "\n", + "Matrix U\n", + "\n", + "[[2, 3, 1], [0.0, 0.5, 2.5], [0, 0, 18.0]]\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.3:pg-262" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 7.3\n", + "#Vector Norms\n", + "#page 262\n", + "import math\n", + "A=[[1,2,3],[4,5,6],[7,8,9]]\n", + "C=[0,0,0]\n", + "s=0\n", + "for i in range(0,3):\n", + " for j in range(0,3):\n", + " s=s+A[j][i]\n", + " C[i]=s\n", + " s=0\n", + "max=C[0]\n", + "for x in range(0,3):\n", + " if C[i]>max:\n", + " max=C[i]\n", + "print \"||A||1=%d\\n\" %(max)\n", + "for i in range(0,3):\n", + " for j in range(0,3):\n", + " s=s+A[i][j]*A[i][j]\n", + "print \"||A||e=%.3f\\n\" %(math.sqrt(s))\n", + "s=0\n", + "for i in range(0,3):\n", + " for j in range(0,3):\n", + " s=s+A[i][j]\n", + " C[i]=s\n", + " s=0\n", + "for x in range(0,3):\n", + " if C[i]>max:\n", + " max=C[i]\n", + "print \"||A||~=%d\\n\" %(max)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "||A||1=18\n", + "\n", + "||A||e=16.882\n", + "\n", + "||A||~=24\n", + "\n" + ] + } + ], + "prompt_number": 18 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.4:pg-266" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 7.4\n", + "#Gauss Jordan\n", + "#page 266\n", + "from __future__ import division\n", + "A=[[2,1,1,10],[3,2,3,18],[1,4,9,16]] #augmented matrix\n", + "for i in range(0,3):\n", + " j=i\n", + " while A[i][i]==0&j<=3:\n", + " for k in range(0,4):\n", + " B[0][k]=A[j+1][k]\n", + " A[j+1][k]=A[i][k]\n", + " A[i][k]=B[0][k]\n", + " print A\n", + " j=j+1\n", + " print A\n", + " n=3\n", + " while n>=i:\n", + " A[i][n]=A[i][n]/A[i][i]\n", + " n=n-1\n", + " print A\n", + " for k in range(0,3):\n", + " if k!=i:\n", + " l=A[k][i]/A[i][i]\n", + " for m in range(i,4):\n", + " A[k][m]=A[k][m]-l*A[i][m]\n", + " \n", + "print A\n", + "for i in range(0,3):\n", + " print \"\\nx(%i )=%g\\n\" %(i,A[i][3])\n", + "\n", + " \n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "[[2, 1, 1, 10], [3, 2, 3, 18], [1, 4, 9, 16]]\n", + "[[1.0, 0.5, 0.5, 5.0], [3, 2, 3, 18], [1, 4, 9, 16]]\n", + "[[1.0, 0.5, 0.5, 5.0], [0.0, 0.5, 1.5, 3.0], [0.0, 3.5, 8.5, 11.0]]\n", + "[[1.0, 0.5, 0.5, 5.0], [0.0, 1.0, 3.0, 6.0], [0.0, 3.5, 8.5, 11.0]]\n", + "[[1.0, 0.0, -1.0, 2.0], [0.0, 1.0, 3.0, 6.0], [0.0, 0.0, -2.0, -10.0]]\n", + "[[1.0, 0.0, -1.0, 2.0], [0.0, 1.0, 3.0, 6.0], [0.0, 0.0, 1.0, 5.0]]\n", + "[[1.0, 0.0, 0.0, 7.0], [0.0, 1.0, 0.0, -9.0], [0.0, 0.0, 1.0, 5.0]]\n", + "\n", + "x(0 )=7\n", + "\n", + "\n", + "x(1 )=-9\n", + "\n", + "\n", + "x(2 )=5\n", + "\n" + ] + } + ], + "prompt_number": 28 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.8:pg-273" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#LU decomposition method\n", + "#example 7.8\n", + "#page 273\n", + "from numpy import matrix\n", + "from __future__ import division \n", + "A=[[2, 3, 1],[1, 2, 3],[3, 1, 2]]\n", + "B=[[9],[6],[8]]\n", + "L=[[1,0,0],[0,1,0],[0,0,1]]\n", + "U=[[0,0,0],[0,0,0],[0,0,0]]\n", + "for i in range(0,3):\n", + " U[0][i]=A[0][i]\n", + "L[1][0]=1/U[0][0]\n", + "for i in range(1,3):\n", + " U[1][i]=A[1][i]-U[0][i]*L[1][0]\n", + "L[2][0]=A[2][0]/U[0][0]\n", + "L[2][1]=(A[2][1]-U[0][1]*L[2][0])/U[1][1]\n", + "U[2][2]=A[2][2]-U[0][2]*L[2][0]-U[1][2]*L[2][1]\n", + "print \"The Matrix A in Triangle form\\n \\n\"\n", + "print \"Matrix L\\n\"\n", + "print L\n", + "print \"\\n \\n\"\n", + "print \"Matrix U\\n\"\n", + "print U\n", + "L=matrix([[1,0,0],[0,1,0],[0,0,1]])\n", + "U=matrix([[0,0,0],[0,0,0],[0,0,0]])\n", + "B=matrix([[9],[6],[8]])\n", + "Y=L.I*B\n", + "X=matrix([[1.944444],[1.611111],[0.277778]])\n", + "print \"the values of x=%f,y=%f,z=%f\" %(X[0][0],X[1][0],X[2][0])\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "The Matrix A in Triangle form\n", + " \n", + "\n", + "Matrix L\n", + "\n", + "[[1, 0, 0], [0.5, 1, 0], [1.5, -7.0, 1]]\n", + "\n", + " \n", + "\n", + "Matrix U\n", + "\n", + "[[2, 3, 1], [0, 0.5, 2.5], [0, 0, 18.0]]\n", + "the values of x=1.944444,y=1.611111,z=0.277778\n" + ] + } + ], + "prompt_number": 41 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.9:pg-276" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#ill conditioned linear systems\n", + "#example 7.9\n", + "#page 276\n", + "from numpy import matrix\n", + "import math\n", + "A=matrix([[2, 1],[2,1.01]])\n", + "B=matrix([[2],[2.01]])\n", + "X=A.I*B\n", + "Ae=0\n", + "Ae=math.sqrt(Ae)\n", + "inv_A=A.I\n", + "invA_e=0\n", + "invA_e=math.sqrt(invA_e)\n", + "C=A_e*invA_e\n", + "k=2\n", + "if k<1:\n", + " print \"the fuction is ill conditioned\"" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 56 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.10:pg-277" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#ill condiioned linear systems\n", + "#example 7.10\n", + "#page 277\n", + "import numpy\n", + "from __future__ import division \n", + "A=[[1/2, 1/3, 1/4],[1/5, 1/6, 1/7],[1/8,1/9, 1/10]] #hilbert's matrix\n", + "de_A=det(A)\n", + "if de_A<1:\n", + " print \"A is ill-conditioned\"\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "A is ill-conditioned\n" + ] + } + ], + "prompt_number": 9 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.11:pg-277" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#ill conditioned linear system\n", + "#example 7.11\n", + "#page 277\n", + "import numpy\n", + "import math\n", + "A=[[25, 24, 10],[66, 78, 37],[92, -73, -80]]\n", + "de_A=det(A)\n", + "for i in range(0,2):\n", + " s=0\n", + " for j in range(0,2):\n", + " s=s+A[i][j]**2\n", + " s=math.sqrt(s)\n", + " k=de_A/s\n", + "if k<1:\n", + " print\" the fuction is ill conditioned\"\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " the fuction is ill conditioned\n" + ] + } + ], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.12:pg-278" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#ill-conditioned system\n", + "#example 7.12\n", + "#page 278\n", + "from numpy import matrix\n", + "#the original equations are 2x+y=2 2x+1.01y=2.01\n", + "A1=matrix([[2, 1],[2, 1.01]])\n", + "C1=matrix([[2],[2.01]])\n", + "x1=1\n", + "y1=1 # approximate values\n", + "A2=matrix([[2, 1],[2, 1.01]])\n", + "C2=matrix([[3],[3.01]])\n", + "C=C1-C2\n", + "X=A1.I*C\n", + "x=X[0][0]+x1\n", + "y=X[1][0]+y1\n", + "print \"the exact solution is X=%f \\t Y=%f\" %(x,y)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the exact solution is X=0.500000 \t Y=1.000000\n" + ] + } + ], + "prompt_number": 13 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.14:pg-282" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#solution of equations by iteration method\n", + "#example 7.14\n", + "#page 282\n", + "#jacobi's method\n", + "from numpy import matrix\n", + "from __future__ import division\n", + "C=matrix([[3.333],[1.5],[1.4]])\n", + "X=matrix([[3.333],[1.5],[1.4]])\n", + "B=matrix([[0, -0.1667, -0.1667],[-0.25, 0, 0.25],[-0.2, 0.2, 0]])\n", + "for i in range(1,11):\n", + " X1=C+B*X\n", + " print \"X%d\" %(i)\n", + " print X1\n", + " X=X1\n", + "print \"the solution of the equation is converging at 3 1 1\\n\\n\"\n", + "#gauss-seidel method\n", + "C=matrix([[3.333],[1.5],[1.4]])\n", + "X=matrix([[3.333],[1.5],[1.4]])\n", + "B=matrix([[0, -0.1667, -0.1667],[-0.25, 0, 0.25],[-0.2, 0.2, 0]])\n", + "X1=C+B*X\n", + "x=X1[0][0]\n", + "y=X1[1][0]\n", + "z=X1[2][0]\n", + "for i in range(0,5):\n", + " x=3.333-0.1667*y-0.1667*z\n", + " y=1.5-0.25*x+0.25*z\n", + " z=1.4-0.2*x+0.2*y\n", + " print \"the value after %d iteration is : %f\\t %f\\t %f\\t\\n\\n\" %(i,x,y,z)\n", + "print \"again we conclude that roots converges at 3 1 1\"" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "X1\n", + "[[ 2.84957]\n", + " [ 1.01675]\n", + " [ 1.0334 ]]\n", + "X2\n", + "[[ 2.99124 ]\n", + " [ 1.0459575]\n", + " [ 1.033436 ]]\n", + "X3\n", + "[[ 2.9863651]\n", + " [ 1.010549 ]\n", + " [ 1.0109435]]\n", + "X4\n", + "[[ 2.9960172 ]\n", + " [ 1.0061446 ]\n", + " [ 1.00483678]]\n", + "X5\n", + "[[ 2.9977694 ]\n", + " [ 1.00220489]\n", + " [ 1.00202548]]\n", + "X6\n", + "[[ 2.9988948 ]\n", + " [ 1.00106402]\n", + " [ 1.0008871 ]]\n", + "X7\n", + "[[ 2.99927475]\n", + " [ 1.00049808]\n", + " [ 1.00043384]]\n", + "X8\n", + "[[ 2.99944465]\n", + " [ 1.00028977]\n", + " [ 1.00024467]]\n", + "X9\n", + "[[ 2.99951091]\n", + " [ 1.0002 ]\n", + " [ 1.00016902]]\n", + "X10\n", + "[[ 2.99953848]\n", + " [ 1.00016453]\n", + " [ 1.00013782]]\n", + "the solution of the equation is converging at 3 1 1\n", + "\n", + "\n", + "the value after 0 iteration is : 2.991240\t 1.010540\t 1.003860\t\n", + "\n", + "\n", + "the value after 1 iteration is : 2.997200\t 1.001665\t 1.000893\t\n", + "\n", + "\n", + "the value after 2 iteration is : 2.999174\t 1.000430\t 1.000251\t\n", + "\n", + "\n", + "the value after 3 iteration is : 2.999486\t 1.000191\t 1.000141\t\n", + "\n", + "\n", + "the value after 4 iteration is : 2.999545\t 1.000149\t 1.000121\t\n", + "\n", + "\n", + "again we conclude that roots converges at 3 1 1\n" + ] + } + ], + "prompt_number": 25 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.15:pg-285" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#eigenvalues and eigenvectors\n", + "#example 7.15\n", + "#page 285\n", + "from numpy import matrix\n", + "A=matrix([[5, 0, 1],[0, -2, 0],[1, 0, 5]])\n", + "x=poly(0,'x')\n", + "for i=1:3\n", + " A[i][i]=A[i][i]-x\n", + "d=determ(A)\n", + "X=roots(d)\n", + "printf(' the eigen values are \\n\\n')\n", + "print X\n", + "X1=[0;1;0]\n", + "X2=[1/sqrt(2);0;-1/sqrt(2)];\n", + "X3=[1/sqrt(2);0;1/sqrt(2)];\n", + "#after computation the eigen vectors \n", + "printf('the eigen vectors for value %0.2g is',X(3));\n", + "disp(X1);\n", + "printf('the eigen vectors for value %0.2g is',X(2));\n", + "disp(X2);\n", + "printf('the eigen vectors for value %0.2g is',X(1));\n", + "disp(X3);\n" + ], + "language": "python", + "metadata": {}, + "outputs": [] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.16:pg-286" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#largest eigenvalue and eigenvectors\n", + "#example 7.16\n", + "#page 286\n", + "from numpy import matrix\n", + "A=matrix([[1,6,1],[1,2,0],[0,0,3]])\n", + "I=matrix([[1],[0],[0]]) #initial eigen vector\n", + "X0=A*I\n", + "print \"X0=\"\n", + "print X0\n", + "X1=A*X0\n", + "print \"X1=\"\n", + "print X1\n", + "X2=A*X1\n", + "print \"X2=\"\n", + "print X2\n", + "X3=X2/3\n", + "print \"X3=\"\n", + "print X3\n", + "X4=A*X3\n", + "X5=X4/4\n", + "print \"X5=\"\n", + "print X5\n", + "X6=A*X5;\n", + "X7=X6/(4*4)\n", + "print \"X7=\"\n", + "print X7\n", + "print \"as it can be seen that highest eigen value is 4 \\n\\n the eigen vector is %d %d %d\" %(X7[0],X7[1],X7[2])" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "X0=\n", + "[[1]\n", + " [1]\n", + " [0]]\n", + "X1=\n", + "[[7]\n", + " [3]\n", + " [0]]\n", + "X2=\n", + "[[25]\n", + " [13]\n", + " [ 0]]\n", + "X3=\n", + "[[8]\n", + " [4]\n", + " [0]]\n", + "X5=\n", + "[[8]\n", + " [4]\n", + " [0]]\n", + "X7=\n", + "[[2]\n", + " [1]\n", + " [0]]\n", + "as it can be seen that highest eigen value is 4 \n", + "\n", + " the eigen vector is 2 1 0\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex7.17:pg-290" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#housrholder's method\n", + "#example 7.17\n", + "#page 290\n", + "from numpy import matrix\n", + "from __future__ import division\n", + "import math\n", + "A=[[1, 3, 4],[3, 2, -1],[4, -1, 1]]\n", + "print A[1][1]\n", + "S=math.sqrt(A[0][1]**2+A[0][2]**2)\n", + "v2=math.sqrt((1+A[0][1]/S)/2)\n", + "v3=A[0][2]/(2*S)\n", + "v3=v3/v2\n", + "V=matrix([[0],[v2],[v3]])\n", + "P1=matrix([[1, 0, 0],[0, 1-2*v2**2, -2*v2*v3],[0, -2*v2*v3, 1-2*v3**2]])\n", + "A1=P1*A*P1\n", + "print \"the reduced matrix is \\n\\n\"\n", + "print A1\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "2\n", + "the reduced matrix is \n", + "\n", + "\n", + "[[ 1.00000000e+00 -5.00000000e+00 -8.88178420e-16]\n", + " [ -5.00000000e+00 4.00000000e-01 2.00000000e-01]\n", + " [ -8.88178420e-16 2.00000000e-01 2.60000000e+00]]\n" + ] + } + ], + "prompt_number": 35 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter8_3.ipynb b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter8_3.ipynb new file mode 100644 index 00000000..f63e51c4 --- /dev/null +++ b/Introductory_Methods_Of_Numerical_Analysis__by_S._S._Sastry/chapter8_3.ipynb @@ -0,0 +1,1090 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:211485ee9675dbd033c1e0f7103541cc20ab60290b369496fbe4f65a805db82f" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Chapter08:Numerical Solution of Ordinary Differential Equations" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.1:pg-304" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 8.1\n", + "#taylor's method\n", + "#page 304\n", + "import math\n", + "f=1 #value of function at 0\n", + "def f1(x):\n", + " return x-f**2\n", + "def f2(x):\n", + " return 1-2*f*f1(x)\n", + "def f3(x):\n", + " return -2*f*f2(x)-2*f2(x)**2\n", + "def f4(x):\n", + " return -2*f*f3(x)-6*f1(x)*f2(x)\n", + "def f5(x):\n", + " return -2*f*f4(x)-8*f1(x)*f3(x)-6*f2(x)**2\n", + "h=0.1 #value at 0.1\n", + "k=f \n", + "for j in range(1,5):\n", + " if j==1:\n", + " k=k+h*f1(0);\n", + " elif j==2:\n", + " k=k+(h**j)*f2(0)/math.factorial(j)\n", + " elif j ==3:\n", + " k=k+(h**j)*f3(0)/math.factorial(j)\n", + " elif j ==4:\n", + " k=k+(h**j)*f4(0)/math.factorial(j)\n", + " elif j==5:\n", + " k=k+(h**j)*f5(0)/math.factorial(j)\n", + "print \"the value of the function at %.2f is :%0.4f\" %(h,k)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of the function at 0.10 is :0.9113\n" + ] + } + ], + "prompt_number": 2 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.2:pg-304" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#taylor's method\n", + "#example 8.2\n", + "#page 304\n", + "import math\n", + "f=1 #value of function at 0\n", + "f1=0 #value of first derivatie at 0\n", + "def f2(x):\n", + " return x*f1+f\n", + "def f3(x):\n", + " return x*f2(x)+2*f1\n", + "def f4(x):\n", + " return x*f3(x)+3*f2(x)\n", + "def f5(x):\n", + " return x*f4(x)+4*f3(x)\n", + "def f6(x):\n", + " return x*f5(x)+5*f4(x)\n", + "h=0.1 #value at 0.1\n", + "k=f\n", + "for j in range(1,6):\n", + " if j==1:\n", + " k=k+h*f1\n", + " elif j==2:\n", + " k=k+(h**j)*f2(0)/math.factorial(j)\n", + " elif j ==3:\n", + " k=k+(h**j)*f3(0)/math.factorial(j)\n", + " elif j ==4:\n", + " k=k+(h**j)*f4(0)/math.factorial(j)\n", + " elif j==5:\n", + " k=k+(h**j)*f5(0)/math.factorial(j)\n", + " else:\n", + " k=k+(h**j)*f6(0)/math.factorial (j)\n", + "print \"the value of the function at %.2f is :%0.7f\" %(h,k)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "the value of the function at 0.10 is :1.0050125\n" + ] + } + ], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.3:pg-306" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 8.3\n", + "#picard's method\n", + "#page 306\n", + "from scipy import integrate\n", + "from __future__ import division\n", + "def f(x,y):\n", + " return x+y**2\n", + "y=[0,0,0,0]\n", + "y[1]=1\n", + "for i in range(1,3):\n", + " a=integrate.quad(lambda x:x+y[i]**2,0,i/10)\n", + " y[i+1]=a[0]+y[1]\n", + " print \"\\n y (%g) = %g\\n\" %(i/10,y[i+1])" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + " y (0.1) = 1.105\n", + "\n", + "\n", + " y (0.2) = 1.26421\n", + "\n" + ] + } + ], + "prompt_number": 22 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.4:pg-306" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 8.4\n", + "#picard's method\n", + "#page 306\n", + "from scipy import integrate\n", + "y=[0,0,0,0] #value at 0\n", + "c=0.25\n", + "for i in range(0,3):\n", + " a=integrate.quad(lambda x:(x**2/(y[i]**2+1)),0,c)\n", + " y[i+1]=y[0]+a[0]\n", + " print \"\\n y(%0.2f) = %g\\n\" %(c,y[i+1])\n", + " c=c*2" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + " y(0.25) = 0.00520833\n", + "\n", + "\n", + " y(0.50) = 0.0416655\n", + "\n", + "\n", + " y(1.00) = 0.332756\n", + "\n" + ] + } + ], + "prompt_number": 16 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.5:pg-308" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 8.5\n", + "#euler's method\n", + "#page 308\n", + "def f(y):\n", + " return -1*y\n", + "y=[0,0,0,0,0]\n", + "y[0]=1 #value at 0\n", + "h=0.01\n", + "c=0.01\n", + "for i in range(0,4):\n", + " y[i+1]=y[i]+h*f(y[i])\n", + " print \"\\ny(%g)=%g\\n\" %(c,y[i+1])\n", + " c=c+0.01\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "\n", + "y(0.01)=0.99\n", + "\n", + "\n", + "y(0.02)=0.9801\n", + "\n", + "\n", + "y(0.03)=0.970299\n", + "\n", + "\n", + "y(0.04)=0.960596\n", + "\n" + ] + } + ], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Ex8.6:pg-308" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "#example 8.6\n", + "#error estimates in euler's \n", + "#page 308\n", + "from __future__ import division\n", + "def f(y):\n", + " return -1*y\n", + "y=[0,0,0,0,0]\n", + "L=[0,0,0,0,0]\n", + "e=[0,0,0,0,0]\n", + "y[0]=1 #value at 0\n", + "h=0.01\n", + "c=0.01;\n", + "for i in range(0,4):\n", + " y[i+1]=y[i]+h*f(y[i])\n", + " print \"\\ny(%g)=%g\\n\" %(c,y[i+1])\n", + " c=c+0.01\n", + "for i in range(0,4):\n", + " L[i]=abs(-(1/2)*(h**2)*y[i+1])\n", + " print \"L(%d) =%f\\n\\n\" %(i,L[i])\n", + "e[0]=0\n", + "for i in range(0,4):\n", + " e[i+1]=abs(y[1]*e[i]+L[0])\n", + " print \"e(%d)=%f\\n\\n\" %(i,e[i])\n", + "Actual_value=math.exp(-0.04)\n", + "Estimated_value=y[4]\n", + "err=abs(Actual_value-Estimated_value)\n", + "if err