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+{
+"cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 13: Numerical Differentiation"
+ ]
+ },
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.1: Differentiation.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.1\n",
+"//Differentiation\n",
+"//Page no. 420\n",
+"clc;close;clear;\n",
+"\n",
+"deff('y=f(x)','y=x^2+5')\n",
+"deff('y=f1(x,h)','y=(f(x+h)-f(x))/h')\n",
+"h=0.01;x=2.4\n",
+"d=f1(x,h)\n",
+"d1=(f1(x+h,h)-f1(x))/h\n",
+"printf('dy\n -- = %g\n dx',d)\n",
+"printf('\n\n\n d2y\n --- = %g\n dx2',d1)"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.2: Calculation_of_x_coordinate_of_Minimum_Point.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.2\n",
+"//Calculation of x-coordinate of Minimum Point \n",
+"//Page no. 422\n",
+"clc;close;clear;\n",
+"\n",
+"for i=1:7\n",
+" for j=1:6\n",
+" z(i,j)=0\n",
+" end\n",
+"end\n",
+"h=0.2\n",
+"printf(' x y d d2 d3 d4\n')\n",
+"printf('--------------------------------------------------------------')\n",
+"for i=1:7\n",
+" z(i,1)=i/5;\n",
+"end\n",
+"z(1,2)=2.10022\n",
+"z(2,2)=1.98730\n",
+"z(3,2)=1.90940\n",
+"z(4,2)=1.86672\n",
+"z(5,2)=1.85937\n",
+"z(6,2)=1.88755\n",
+"z(7,2)=1.95147\n",
+"for i=3:6\n",
+" for j=1:9-i\n",
+" z(j,i)=z(j+1,i-1)-z(j,i-1)\n",
+" end\n",
+"end\n",
+"disp(z)\n",
+"\n",
+"s=poly(0,'s')\n",
+"p=z(5,2);k=4;\n",
+"for i=3:5\n",
+" r=1;\n",
+" for j=1:i-2\n",
+" r=r*(s+(j-1))\n",
+" end\n",
+" r=r*z(k,i)/factorial(j);\n",
+" k=k-1;\n",
+" p=p+r;\n",
+" \n",
+"end\n",
+"disp(p)\n",
+"s=(-z(4,3)+z(3,4)/2)/z(3,4)\n",
+"disp(s,'s=')\n",
+"x=z(5,1)+s*h\n",
+"disp(x,'x=')"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.3: Newton_Forward_Difference_Formula.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.3\n",
+"//Newton's Forward Difference Formula\n",
+"//Page no. 423\n",
+"clc;close;clear;\n",
+"printf(' x\t\t y\t\t d\t\t d2\t\t d3\t\t d4\n')\n",
+"printf('------------------------------------------------------------------------------------------')\n",
+"h=0.05;\n",
+"z=[1.00,1.00000;1.05,1.02470;1.10,1.04881;1.15,1.07238;1.20,1.09544;1.25,1.11803;1.30,1.14018]\n",
+"deff('y=f1(x,s)','y=(z(x,3)+(s-1/2)*z(x,4)+z(x,5)*(3*s^2-6*s+2)/6)/h')\n",
+"deff('y=f2(x,s)','y=(z(x,4)+z(x,5)*(s-1))/h^2')\n",
+"deff('y=f3(x,s)','y=z(x,5)/h^3')\n",
+"for i=3:6\n",
+" for j=1:9-i\n",
+" z(j,i)=z(j+1,i-1)-z(j,i-1)\n",
+" end\n",
+"end\n",
+"printf('\n')\n",
+"for i=1:7\n",
+" for j=1:6\n",
+" if z(i,j)==0 then\n",
+" printf(' \t')\n",
+" else\n",
+" printf('%.7f\t',z(i,j))\n",
+" end\n",
+" end\n",
+" printf('\n')\n",
+"end\n",
+"s=poly(0,'s')\n",
+"p=z(5,2);k=4;\n",
+"for i=3:5\n",
+" r=1;\n",
+" for j=1:i-2\n",
+" r=r*(s+(j-1))\n",
+" end\n",
+" r=r*z(k,i)/factorial(j);\n",
+" k=k-1;\n",
+" p=p+r;\n",
+" \n",
+"end\n",
+"disp(p,'y(s) = ')\n",
+"printf('\n\ny1(1) = %g',f1(1,0))\n",
+"printf('\n\ny2(1) = %g',f2(1,0))\n",
+"printf('\n\ny3(1) = %g',f3(1,0))\n",
+"printf('\n\ny1(1.025) = %g',f1(1,0.5))"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.4: Newton_Backward_Difference_Formula.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.4\n",
+"//Newton's Backward Difference Formula\n",
+"//Page no. 425\n",
+"clc;close;clear;\n",
+"printf(' x\t\t y\t\t d\t\t d2\t\t d3\t\t d4\n')\n",
+"printf('------------------------------------------------------------------------------------------')\n",
+"h=0.02;\n",
+"z=[0.96,1.8025;0.98,1.7939;1.00,1.7851;1.02,1.7763;1.04,1.7673];\n",
+"deff('y=f1(x,s)','y=(z(x,3)+(s+1/2)*z(x,4))/h')\n",
+"for i=3:6\n",
+" for j=1:7-i\n",
+" z(j,i)=z(j+1,i-1)-z(j,i-1)\n",
+" end\n",
+"end\n",
+"printf('\n')\n",
+"for i=1:5\n",
+" for j=1:6\n",
+" if z(i,j)==0 then\n",
+" printf(' \t')\n",
+" else\n",
+" printf('%.7f\t',z(i,j))\n",
+" end\n",
+" end\n",
+" printf('\n')\n",
+"end\n",
+"printf('\n\ny1(1) = %g',f1(2,0))\n",
+"printf('\n\ny1(1.03) = %g',f1(4,0.5))"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.5: Stirlings_Central_Difference_Derivatives.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.5\n",
+"//Stirlings Central Difference Derivatives\n",
+"//Page no. 426\n",
+"clc;close;clear;\n",
+"printf(' x\t\t y\t\t d\t\t d2\t\t d3\n')\n",
+"printf('---------------------------------------------------------------------------')\n",
+"h=0.01;s=0.5;\n",
+"deff('y=f1(x,s)','y=((z(x,3)+z(x-1,3))/2+s*z(x-1,4)+(z(x-1,5)+z(x-2,5))*(3*s^2-1)/12)/h')\n",
+"deff('y=f2(x,s)','y=(z(x-1,4))/h^2')\n",
+"deff('y=f3(x,s)','y=(z(x-1,5)+z(x-2,5))/(2*h^3)')\n",
+"z=[1.00,1.00000;1.01,1.00499;1.02,1.00995;1.03,1.01489;1.04,1.01980;1.05,1.02470;1.06,1.02956;1.07,1.03441;1.08,1.03923;1.09,1.04403;1.10,1.04881;1.11,1.05357;1.12,1.05830;1.13,1.06301;1.14,1.06771;1.15,1.07238;1.16,1.07703];\n",
+"for i=3:5\n",
+" for j=1:19-i\n",
+" z(j,i)=z(j+1,i-1)-z(j,i-1)\n",
+" end\n",
+"end\n",
+"printf('\n')\n",
+"for i=1:17\n",
+" for j=1:5\n",
+" if z(i,j)==0 then\n",
+" printf(' \t')\n",
+" else\n",
+" printf('%.7f\t',z(i,j))\n",
+" end\n",
+" end\n",
+" printf('\n')\n",
+"end\n",
+"printf('\n\ny1(1.125) = %g (exact value = 0.4771404)',f1(13,0.5))\n",
+"printf('\n\ny2(1.125) = %g (exact value = -0.20951)',f2(13,0.5))\n",
+"printf('\n\ny3(1.125) = %g (exact value = 0.27935)',f3(13,0.5))"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.6: Extrapolatio.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.6\n",
+"//Extrapolation\n",
+"//Page no. 430\n",
+"clc;close;clear;\n",
+"x=[-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8];\n",
+"y=[0.2019,0.30119,0.44933,0.67032,1,1.49182,2.22554,3.32012,4.95303]\n",
+"for i=1:4\n",
+" printf('\nh = %g\n',x(10-i))\n",
+" y1=(y(10-i)-y(i))/(2*x(10-i))\n",
+" printf('f1(0) = %g\n\n',y1)\n",
+"end"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.7: Richardson_Extrapolation.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.7\n",
+"//Richardson Extrapolation\n",
+"//Page no. 431\n",
+"clc;close;clear;\n",
+"\n",
+"deff('y=f(x)','y=exp(2*x)')\n",
+"e=10^-4;h=0.8;\n",
+"D1=0;\n",
+"for i=1:4\n",
+" printf('\n')\n",
+" for j=1:i\n",
+" if j==1 then\n",
+" D(i,j)=(f(h)-f(-h))/(2*h)\n",
+" else\n",
+" D(i,j)=D(i,j-1)+(D(i,j-1)-D(i-1,j-1))/(2^(2*(j-1))-1)\n",
+" end\n",
+" printf('%g\t\t',D(i,j))\n",
+" end\n",
+" h=h/2\n",
+"end\n",
+"printf('\n\n\t\t\t\t\t\t 2x\nHence, the derivative of the function y = f(x) = e at x=0 is D(3,3) = %g',D(i,j))"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 13.8: Applicatio.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Example 13.8\n",
+"//Application\n",
+"//Page no. 433\n",
+"clc;close;clear;\n",
+"\n",
+"deff('y=f(x)','y=2/x^2')\n",
+"a=1;b=2;a1=1;b1=0;\n",
+"N=4;\n",
+"h=(b-a)/(N+1);\n",
+"for j=1:N\n",
+" s(j)=f(a+j*h)\n",
+"end\n",
+"for i=1:N\n",
+" for j=1:N\n",
+" if abs(i-j)==1 then\n",
+" A(i,j)=-1\n",
+" end\n",
+" if i==j then\n",
+" A(i,j)=2+s(i)*h^2\n",
+" end\n",
+" end\n",
+" if i==1 then\n",
+" k(i,1)=s(i)+a1/h^2\n",
+" elseif i==N\n",
+" k(i,1)=s(i)+b1/h^2\n",
+" else\n",
+" k(i,1)=s(i)\n",
+" end\n",
+"end\n",
+"disp(A,'A = ')\n",
+"disp(k,'k = ')"
+ ]
+ }
+],
+"metadata": {
+ "kernelspec": {
+ "display_name": "Scilab",
+ "language": "scilab",
+ "name": "scilab"
+ },
+ "language_info": {
+ "file_extension": ".sce",
+ "help_links": [
+ {
+ "text": "MetaKernel Magics",
+ "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+ }
+ ],
+ "mimetype": "text/x-octave",
+ "name": "scilab",
+ "version": "0.7.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
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