高斯近似

1.(a) fx= x2

step 1: calculate qi

q0=a0-<a0,q0>q0=1;

q1=a1-<a1,q0>q0=32x-1;

step 2: calculate ci

c0=<f,q0>=01fq0dx=13;

c1=<f,q1>=01fq1dx=36;

step 3: calculate p(x)

px=c0q0+c1q1=x-16;

step 4: get result

r=min01(fx-p(x))2dx= 1180≈0.00555….

Fig 1. f(x) = x^2

 

(b) fx= x3/2

step 1: calculate qi

q0=a0=1;

q1=a1-<a1,q0>q0=32x-1;

q2=a2-<a2,q0>q0-<a2,q1>q1

=180x2-x+16;

step 2: calculate ci

c0=<f,q0>=01fq0dx=25;

c1=<f,q1>=01fq1dx=6335;

c2=<f,q2>=01fq2dx=25105;

step 3: calculate p(x)

px=c0q0+c1q1=47x2+1635x-2105;

step 4: get result

r=min01(fx-p(x))2dx= 144100≈2.26e-5.

Fig 2. f(x) = x^(3/2)

(c) Program in Matlab to Calculate qi, ci and p(x), then get the figure as follow

 

Fig 3. f(x) = 1/(1+x^2)