{
"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 = ')"
   ]
   }
],
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		   "help_links": [
			{
			 "text": "MetaKernel Magics",
			 "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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