{
"cells": [
 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 14: Numerical Integration"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.10: Spline_Integration_Method.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.10\n",
"//Spline Integration Method\n",
"//Page no. 478\n",
"clc;close;clear;\n",
"\n",
"deff('y=f(x)','y=sind(%pi*x)')\n",
"deff('y=f1(x,h)','y=(f(x+h)-f(x))/h')\n",
"h=0.01;\n",
"n=2;h=0.5;a=0;b=1;\n",
"disp(integrate('f(x)','x',0,1),'I = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.11: Trapezoidal_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.1\n",
"//Trapezoidal Rule\n",
"//Page no 440\n",
"clc;clear;close;\n",
"x1=1.46\n",
"for i=1:6\n",
"    x(1,i)=x1+i/100\n",
"end\n",
"y=[3.86,3.90,3.96,4.02,4.06,4.12]\n",
"\n",
"//trapezoidal rule\n",
"S=0;\n",
"h=(x(6)-x1)/6\n",
"for i=1:6\n",
"    if(i==1 | i==6)\n",
"        S=S+y(i)\n",
"    else\n",
"        S=S+2*y(i)\n",
"    end\n",
"end\n",
"S=S*h/2\n",
"printf('\n I = %g',S)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.14: Trapezoidal_and_Simpsons_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.14\n",
"//Trapezoidal and Simpsons Rule\n",
"//Page no. 486\n",
"clc;close;clear;\n",
"\n",
"x(1)=0.5;y(1)=0.5;h=0.25\n",
"for i=2:3\n",
"    x(i)=x(i-1)+h\n",
"    y(i)=y(i-1)+h\n",
"end\n",
"printf(' y/x\t|\t%g\t%g\t%g',x(1),x(2),x(3))\n",
"printf('\n--------|---------------------------')\n",
"for i=1:3\n",
"     printf('\n%g\t|\t',y(i))\n",
"    for j=1:3\n",
"        z(i,j)=x(j)*y(i)\n",
"        printf('%g\t',z(i,j))\n",
"    end\n",
"end\n",
"\n",
"//trapezoidal rule\n",
"s=0;\n",
"for i=1:3\n",
"    for j=1:3\n",
"        if i==1 & j==1 then\n",
"            s=s+z(i,j)\n",
"        elseif i==3 & j==3\n",
"            s=s+z(i,j)\n",
"        else\n",
"            s=s+2*z(i,j)\n",
"        end\n",
"    end\n",
"end\n",
"s=(s*(h^2))/4\n",
"printf('\n\n')\n",
"disp(s,'Trapezoidal Rule Sum = ')\n",
"printf('\n\n')\n",
"//simpsons rule\n",
"s=0;\n",
"for i=1:3\n",
"    for j=1:3\n",
"        if i/2-int(i/2)==0 & j/2-int(j/2)==0 then\n",
"            s=s+16*z(i,j)\n",
"        elseif i/2-int(i/2)~=0 & j/2-int(j/2)~=0\n",
"            s=s+z(i,j)\n",
"        else\n",
"            s=s+4*z(i,j)\n",
"        end\n",
"    end\n",
"end\n",
"s=(s*(h^2))/9\n",
"disp(s,'Simpsons Rule Sum = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.15: Trapezoidal_and_Simpsons_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.15\n",
"//Trapezoidal and Simpsons Rule\n",
"//Page no. 487\n",
"clc;close;clear;\n",
"\n",
"x(1)=0;y(1)=0;h=0.25\n",
"for i=2:5\n",
"    x(i)=x(i-1)+h\n",
"    y(i)=y(i-1)+h\n",
"end\n",
"printf(' y/x\t|\t%g\t%g\t%g\t%g\t%g',x(1),x(2),x(3),x(4),x(5))\n",
"printf('\n--------|-------------------------------------------')\n",
"for i=1:5\n",
"     printf('\n%g\t|\t',y(i))\n",
"    for j=1:5\n",
"        z(i,j)=x(j)*y(i)\n",
"        printf('%g\t',z(i,j))\n",
"    end\n",
"end\n",
"\n",
"//trapezoidal rule\n",
"s=0;\n",
"for i=1:5\n",
"    for j=1:5\n",
"        if i==1 & j==1 then\n",
"            s=s+z(i,j)\n",
"        elseif i==5 & j==5\n",
"            s=s+z(i,j)\n",
"        else\n",
"            s=s+2*z(i,j)\n",
"        end\n",
"    end\n",
"end\n",
"s=(s*(h^2))/4\n",
"printf('\n\n')\n",
"disp(s,'Trapezoidal Rule Sum = ')\n",
"printf('\n\n')\n",
"\n",
"//simpsons rule\n",
"s=0;\n",
"for i=1:5\n",
"    for j=1:5\n",
"        if i/2-int(i/2)==0 & j/2-int(j/2)==0 then\n",
"            if i==j then\n",
"                s=s+16*z(i,j)\n",
"            else\n",
"                s=s+4*z(i,j)\n",
"            end\n",
"            \n",
"        elseif i/2-int(i/2)~=0 & j/2-int(j/2)~=0\n",
"            s=s+z(i,j)\n",
"        else\n",
"            s=s+4*z(i,j)\n",
"        end\n",
"    end\n",
"end\n",
"s=(s*(h^2))/9\n",
"disp(s,'Simpsons Rule Sum = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.16: Multiple_Integration_with_Variable_Limits.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.16\n",
"//Multiple Integration with Variable Limits\n",
"//Page no. 491\n",
"clc;close;clear;\n",
"\n",
"deff('z=f(x)','z=x+1')\n",
"deff('z=f1(y)','z=(y+1)^3*(y+3)^2')\n",
"s=5/9*f(-sqrt(3/5))+8/9*f(0)+5/9*f(sqrt(3/5))\n",
"s=s*5/9*f1(-sqrt(3/5))+8/9*f1(0)+5/9*f1(sqrt(3/5))\n",
"s=s/256;\n",
"disp(s,'I = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.18: Integration.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.18\n",
"//Integration\n",
"//Page no. 494\n",
"clc;close;clear;\n",
"\n",
"s=integrate('x^2*sin(x^2)','x',0,1)\n",
"disp(s,'I = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.19: Integration.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.19\n",
"//Integration\n",
"//Page no. 494\n",
"clc;close;clear;\n",
"\n",
"s=integrate('sin(t)/t','t',1,999)\n",
"disp(s,'I = ')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.2: Simpsons_1_3rd_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.2\n",
"//Simpsons 1/3rd Rule\n",
"//Page no 442\n",
"clc;clear;close;\n",
"x(1,1)=0\n",
"for i=2:9\n",
"\n",
"    x(1,i)=x(1,1)+(i-1)*10\n",
"end\n",
"y=[30,31.63,33.44,35.47,37.75,40.33,43.29,46.69,50.67]\n",
"\n",
"//trapezoidal rule\n",
"S=0;\n",
"h=(x(9)-x(1))/8\n",
"for j=1:9\n",
"    S=0;\n",
"    for i=1:j\n",
"        if(i==1 | i==j)\n",
"            S=S+y(i)\n",
"        else\n",
"            S=S+2*y(i)\n",
"        end\n",
"    end\n",
"    S=S*h/2\n",
"    printf('\n Velocity at t (%i) = %.2f',x(j),S)\n",
"    y1(j)=S\n",
"end\n",
"\n",
"y1(1)=0;\n",
"//Simpsons 1/3rd Rule\n",
"S=0;\n",
"h=(x(9)-x(1))/8\n",
"for i=1:9\n",
"    if(i==1 | i==9)\n",
"        S=S+y1(i)\n",
"    elseif(((i)/2)-fix((i)/2)==0)\n",
"        S=S+4*y1(i)\n",
"    else\n",
"        S=S+2*y1(i)\n",
"    end\n",
"end\n",
"S=S*h/3;\n",
"S=S/1000\n",
"printf('\n\nSimpsons 1/3rd Rule Sum = %g km',S)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.3: Trapezoidal_Rule_and_Simpsons_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.3\n",
"//Trapezoidal Rule and Simpsons Rule\n",
"//Page no. 442\n",
"clc;close;clear;\n",
"n=2;a=0;b=1;\n",
"h=(b-a)/n\n",
"deff('y=f(x)','y=1/(1+x)')\n",
"for i=0:2\n",
"    x(i+1)=i/2;\n",
"    y(i+1)=f(x(i+1))\n",
"end\n",
"printf('xi\t ')\n",
"for i=1:3\n",
"    printf('%g\t ',x(i))\n",
"end\n",
"printf('\n yi\t')\n",
"for i=1:3\n",
"    printf('1/%g\t',1+(i-1)/2)\n",
"end\n",
"\n",
"//trapezoidal rule\n",
"S=0;\n",
"for i=1:3\n",
"    if(i==1 | i==3)\n",
"        S=S+y(i)\n",
"    else\n",
"        S=S+2*y(i)\n",
"    end\n",
"end\n",
"S=S*h/2\n",
"printf('\n\nTrapezoidal Rule Sum = %g',S)\n",
"\n",
"//Simpsons 1/3rd Rule\n",
"S=0;\n",
"for i=1:3\n",
"    if(i==1 | i==3)\n",
"        S=S+y(i)\n",
"    elseif(((i)/2)-fix((i)/2)==0)\n",
"        S=S+4*y(i)\n",
"    else\n",
"        S=S+2*y(i)\n",
"    end\n",
"end\n",
"S=S*h/3\n",
"printf('\n\nSimpsons 1/3rd Rule Sum = %g',S)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.5: Romberg_Method.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.5\n",
"//Romberg Method\n",
"//Page no. 457\n",
"clc;close;clear;\n",
"\n",
"deff('y=f(x)','y=1/(1+x)')\n",
"\n",
"h=[0.5,0.25,0.125]\n",
"for k=1:3\n",
"    for i=0:h(k):1\n",
"        x(i/h(k)+1)=i;\n",
"        y(i/h(k)+1)=f(x(i/h(k)+1))\n",
"    end\n",
"    n=1+(1/h(k))\n",
"    //trapezoidal rule\n",
"    S=0;\n",
"     for i=1:n\n",
"        if(i==1 | i==n)\n",
"            S=S+y(i)\n",
"        else\n",
"            S=S+2*y(i)\n",
"        end\n",
"    end\n",
"    S=S*h(k)/2\n",
"    printf('\n\nI(%g) = %g',h(k),S)\n",
"    z(2*k-1,1)=S\n",
"end\n",
"for i=2:3\n",
"    for k=1:4-i\n",
"    z(k*2+i-2,i)=z(2*k-1+i,i-1)+(z(2*k-1+i,i-1)-z(2*k-3+i,i-1))/3\n",
"end\n",
"end\n",
"\n",
"printf('\n\n')\n",
"disp(z,'The Table of values:')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.7: Gaussian_Quadrature_Formula.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.7\n",
"//Gaussian Quadrature Formula\n",
"//Page no. 463\n",
"clc;close;clear;\n",
"\n",
"deff('y=f(x)','y=cos(x)*log(x)')\n",
"s=0;\n",
"for i=0:2:2000\n",
"    s=s+integrate('((-1)^(i/2))*(x^i)/factorial(i)*log(x)','x',0,1)\n",
"end\n",
"disp(s,'Till 1000 terms .... I =')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.8: Gauss_Legendre_Two_Point_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.8\n",
"//Gauss Legendre Two Point Rule\n",
"//Page no. 472\n",
"clc;close;clear;\n",
"\n",
"deff('y=f(x)','y=1/(x+3)')\n",
"s=integrate('f(x)','x',-1,1)\n",
"printf('By Direct Method, I = %g',s)\n",
"s=f(-1/sqrt(3))+f(1/sqrt(3))\n",
"printf('\n\n By Gauss-Legendre 2 point rule, I = %g',s)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 14.9: Gauss_Legendre_Three_Point_Rule.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//Example 14.9\n",
"//Gauss Legendre Three Point Rule\n",
"//Page no. 473\n",
"clc;close;clear;\n",
"\n",
"deff('y=f(x)','y=1/(x+3)')\n",
"s=integrate('f(x)','x',-1,1)\n",
"printf('By Direct Method, I = %g',s)\n",
"s=5/9*f(-sqrt(3/5))+8/9*f(0)+5/9*f(sqrt(3/5))\n",
"printf('\n\n By Gauss-Legendre 3 point rule, I = %g',s)"
   ]
   }
],
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		   "language": "scilab",
		   "name": "scilab"
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		   "file_extension": ".sce",
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			{
			 "text": "MetaKernel Magics",
			 "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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