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-rwxr-xr-x75/CH4/EX4.1/ex_12.sce8
-rwxr-xr-x75/CH4/EX4.10/ex_9.sce9
-rwxr-xr-x75/CH4/EX4.11/ex_11.sce25
-rwxr-xr-x75/CH4/EX4.2/ex_1.sce60
-rwxr-xr-x75/CH4/EX4.3/ex_2.sce23
-rwxr-xr-x75/CH4/EX4.4/ex_3.sce12
-rwxr-xr-x75/CH4/EX4.5/ex_5.sce9
-rwxr-xr-x75/CH4/EX4.6/ex_5.sce8
-rwxr-xr-x75/CH4/EX4.7/ex_6.sce27
-rwxr-xr-x75/CH4/EX4.8/ex_7.sce28
-rwxr-xr-x75/CH4/EX4.9/ex_8.sce20
11 files changed, 229 insertions, 0 deletions
diff --git a/75/CH4/EX4.1/ex_12.sce b/75/CH4/EX4.1/ex_12.sce
new file mode 100755
index 000000000..647febdb9
--- /dev/null
+++ b/75/CH4/EX4.1/ex_12.sce
@@ -0,0 +1,8 @@
+ // PG (199)
+
+x = poly(0,"x");
+p3 = 1 + x + (1/2)*x^(2) + (1/6)*x^3
+deff('[y]=f(x)','y=exp(x)')
+funcprot(0)
+x = -1:0.01:1;
+f(x) - p3
diff --git a/75/CH4/EX4.10/ex_9.sce b/75/CH4/EX4.10/ex_9.sce
new file mode 100755
index 000000000..b3ba594e2
--- /dev/null
+++ b/75/CH4/EX4.10/ex_9.sce
@@ -0,0 +1,9 @@
+ // PG (227)
+
+deff('[y]=f(x)','y=exp(x)')
+c3=0.994571+0.997308*x+0.542991*x^2+0.177347*x^3;
+norm(exp(x)-c3,'inf')
+
+// as obtained in the example 6, c4 = 0.00547, T4(x) = (-1)
+// c4*T4(x) = 0.00547 * (-1)
+// norm(exp(x)-q3,'inf') = 0.00553 \ No newline at end of file
diff --git a/75/CH4/EX4.11/ex_11.sce b/75/CH4/EX4.11/ex_11.sce
new file mode 100755
index 000000000..2ac7242f1
--- /dev/null
+++ b/75/CH4/EX4.11/ex_11.sce
@@ -0,0 +1,25 @@
+ // PG (234)
+
+deff('[y]=f(x)','y=exp(x)')
+x = -1:0.01:1;
+// For
+n = 1;
+x = [-1 0 1];
+E1 = 0.272;
+F1 = 1.2715 + 1.1752*x;
+
+// Relative errors
+
+x = -1.0;
+exp(x) - F1;
+r1 = ans(1,1)
+x = 0.1614;
+exp(x) - F1;
+r2 = ans(1,2)
+x = 1.0;
+exp(x) - F1;
+r3 = ans(1,3)
+
+F3 = 0.994526 + 0.995682*x + 0.543981*x*x + 0.179519*x*x*x;
+x = [-1.0 -0.6832 0.0493 0.7324 1.0]
+exp(x) - F3 // relative errors \ No newline at end of file
diff --git a/75/CH4/EX4.2/ex_1.sce b/75/CH4/EX4.2/ex_1.sce
new file mode 100755
index 000000000..d0d5ebc5f
--- /dev/null
+++ b/75/CH4/EX4.2/ex_1.sce
@@ -0,0 +1,60 @@
+ // PG (200)
+
+deff('[y]=f(x)','y=exp(x)')
+
+xset('window',0);
+x=-1:.01:1; // defining the range of x.
+y=feval(x,f);
+
+a=gca();
+
+a.y_location = "origin";
+
+a.x_location = "origin";
+plot(x,y) // instruction to plot the graph
+
+
+
+// possible approximation
+// y = q1(x)
+
+// Let e(x) = exp(x) - [a0+a1*x]
+// q1(x) & exp(x) must be equal at two points in [-1,1], say at x1 & x2
+// sigma1 = max(abs(e(x)))
+// e(x1) = e(x2) = 0.
+// By another argument based on shifting the graph of y = q1(x),
+// we conclude that the maximum error sigma1 is attained at exactly 3 points.
+// e(-1) = sigma1
+// e(1) = sigma1
+// e(x3) = -sigma1
+// x1 < x3 < x2
+// Since e(x) has a relative minimum at x3, we have e'(x) = 0
+// Combining these 4 equations, we have..
+// exp(-1) - [a0-a1] = sigma1 ------------------(i)
+// exp(1) - [a0+a1] = p1 -----------------------(ii)
+// exp(x3) - [a0+a1*x3] = -sigma1 --------------(iii)
+// exp(x3) - a1 = 0 ----------------------------(iv)
+
+// These have the solution
+
+a1 = (exp(1) - exp(-1))/2
+x3 = log(a1)
+sigma1 = 0.5*exp(-1) + x3*(exp(1) - exp(-1))/4
+a0 = sigma1 + (1-x3)*a1
+
+x = poly(0,"x");
+// Thus,
+q1 = a0 + a1*x
+
+deff('[y1]=f(x)','y1=1.2643+1.1752*x')
+
+xset('window',0);
+x=-1:.01:1; // defining the range of x.
+y=feval(x,f);
+
+a=gca();
+
+a.y_location = "origin";
+
+a.x_location = "origin";
+plot(x,y) // instruction to plot the graph
diff --git a/75/CH4/EX4.3/ex_2.sce b/75/CH4/EX4.3/ex_2.sce
new file mode 100755
index 000000000..20950fb4b
--- /dev/null
+++ b/75/CH4/EX4.3/ex_2.sce
@@ -0,0 +1,23 @@
+ // PG (205)
+
+deff('[y]=f(x)','y=exp(x)')
+
+x=-1:.01:1; // defining the range of x
+
+// Let r1(x) = b0 + b1(x)
+// Minimize
+// ||f-r1||^2 = integrate('(exp(x)-b0-b1*x)^2','x',-1,1) = F(b0,b1)
+// F = integrate('exp(2*x) + b0^2 + (b1^2)*(x^2) - 2*b0*x*exp(x) + 2*b0*b1*x','x',b0,b1)
+// To find a minimum, we set
+
+// df/db0 = 0
+// df/db1 = 0-----------necessary conditions at a minimal point
+// On solving, we get the values of b0 & b1
+
+b0 = 0.5*integrate('exp(x)','x',-1,1)
+b1 = 1.5*integrate('x*exp(x)','x',-1,1)
+r1 = b0+b1*x;
+norm(exp(x)-r1,'inf') // least squares approximation
+
+r3 = 0.996294 + 0.997955*x + 0.536722*x^2 + 0.176139*x^3
+norm(exp(x)-r3,'inf') // cubic least squares approximation \ No newline at end of file
diff --git a/75/CH4/EX4.4/ex_3.sce b/75/CH4/EX4.4/ex_3.sce
new file mode 100755
index 000000000..59ed524c3
--- /dev/null
+++ b/75/CH4/EX4.4/ex_3.sce
@@ -0,0 +1,12 @@
+ // PG (206)
+
+// The following are the weight functions of most interest in the
+// developments of this text:
+
+// w(x)=1 a < = x < = b
+
+// w(x)=1/sqrt(1-x^2) -1 < = x < = 1
+
+// w(x)=exp(-x) 0 < = x < infinity
+
+// w(x)=exp(-x^2) -infinity < x < infinity \ No newline at end of file
diff --git a/75/CH4/EX4.5/ex_5.sce b/75/CH4/EX4.5/ex_5.sce
new file mode 100755
index 000000000..a52e70ef8
--- /dev/null
+++ b/75/CH4/EX4.5/ex_5.sce
@@ -0,0 +1,9 @@
+ // PG (215)
+
+// for laguerre polynomials,
+// L(n+1)=1*[2*n+1-x]*L(n)/(n+1) - n*L(n-1)/(n+1)
+
+// for Legendre polynomials,
+
+// P(n+1)= (2*n +1)*x*P(n)/(n+1) - n*P(n-1)/(n+1)
+
diff --git a/75/CH4/EX4.6/ex_5.sce b/75/CH4/EX4.6/ex_5.sce
new file mode 100755
index 000000000..61e424105
--- /dev/null
+++ b/75/CH4/EX4.6/ex_5.sce
@@ -0,0 +1,8 @@
+ // PG (215)
+
+// for laguerre polynomials,
+// L(n+1)=1*[2*n+1-x]*L(n)/(n+1) - n*L(n-1)/(n+1)
+
+// for Legendre polynomials,
+
+// P(n+1)= (2*n +1)*x*P(n)/(n+1) - n*P(n-1)/(n+1) \ No newline at end of file
diff --git a/75/CH4/EX4.7/ex_6.sce b/75/CH4/EX4.7/ex_6.sce
new file mode 100755
index 000000000..b87b04d2a
--- /dev/null
+++ b/75/CH4/EX4.7/ex_6.sce
@@ -0,0 +1,27 @@
+ // PG (219)
+
+deff('[y]=f(x)','y=exp(x)')
+
+x=-1:.01:1; // defining the range of x
+
+// Let r1(x) = b0 + b1(x)
+// Minimize
+// ||f-r1||^2 = integrate('(exp(x)-b0-b1*x)^2','x',-1,1) = F(b0,b1)
+// F = integrate('exp(2*x) + b0^2 + (b1^2)*(x^2) - 2*b0*x*exp(x) + 2*b0*b1*x','x',b0,b1)
+// To find a minimum, we set
+
+// df/db0 = 0
+// df/db1 = 0-----------necessary conditions at a minimal point
+// On solving, we get the values of b0 & b1
+
+b0 = 0.5*integrate('exp(x)','x',-1,1);
+b1 = 1.5*integrate('x*exp(x)','x',-1,1);
+r1 = b0+b1*x;
+norm(exp(x)-r1,'inf'); // least squares approximation
+
+r3 = 0.996294 + 0.997955*x + 0.536722*x^2 + 0.176139*x^3;
+norm(exp(x)-r3,'inf'); // cubic least squares approximation
+
+// average error E
+
+E = norm(exp(x)-r3,2)/sqrt(2)
diff --git a/75/CH4/EX4.8/ex_7.sce b/75/CH4/EX4.8/ex_7.sce
new file mode 100755
index 000000000..01c0b0149
--- /dev/null
+++ b/75/CH4/EX4.8/ex_7.sce
@@ -0,0 +1,28 @@
+ // PG (220)
+
+deff('[y]=f(x)','y=exp(x)')
+
+// Chebyshev expansion coefficients for exp(x)
+// j = 0
+C0=2*(integrate('exp(cos(x))','x',0,3.14))/(3.14)
+
+// j = 1
+C1=2*(integrate('exp(cos(x))*cos(x)','x',0,3.14))/(3.14)
+
+// j = 2
+C2=2*(integrate('exp(cos(x))*cos(2*x)','x',0,3.14))/(3.14)
+
+// j = 3
+C3=2*(integrate('exp(cos(x))*cos(3*x)','x',0,3.14))/(3.14)
+
+// j = 4
+C4=2*(integrate('exp(cos(x))*cos(4*x)','x',0,3.14))/(3.14)
+
+// j = 5
+C5=2*(integrate('exp(cos(x))*cos(5*x)','x',0,3.14))/(3.14)
+
+// we obtain
+c1=1.266+1.130*x;
+c3=0.994571+0.997308*x+0.542991*x^2+0.177347*x^3;
+norm(exp(x)-c1,'inf')
+norm(exp(x)-c3,'inf') \ No newline at end of file
diff --git a/75/CH4/EX4.9/ex_8.sce b/75/CH4/EX4.9/ex_8.sce
new file mode 100755
index 000000000..7c502a299
--- /dev/null
+++ b/75/CH4/EX4.9/ex_8.sce
@@ -0,0 +1,20 @@
+ // PG (223)
+
+deff('[y]=f(x)','y=exp(x)')
+
+x=[-1.0 -0.6919 0.0310 0.7229 1.0]; // defining x
+
+r3 = 0.996294 + 0.997955*x + 0.536722*x^2 + 0.176139*x^3;
+norm(exp(x)-r3,'inf'); // cubic least squares approximation
+deff('[y]=g(x)','y=0.994571+0.997308*x+0.542991*x^2+0.177347*x^3')
+// c3=g(x);
+x1=x(1,1);
+(exp(x1)-g(x1))
+x2=x(1,2);
+(exp(x2)-g(x2))
+x3=x(1,3);
+(exp(x3)-g(x3))
+x4=x(1,4);
+(exp(x4)-g(x4))
+x5=x(1,5);
+(exp(x5)-g(x5))
generated by cgit (git 2.34.1) at 2024-12-20 12:02:02 +0000