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// 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
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