﻿ 17074227-作业8 - 计算数学达人 - 专，学者，数值代数，微分方程数值解

# 4.3Exercises

1. Apply classical Gram–Schmidt orthogonalization to nd the full QR factorization of the following matrices:

b -4   -4

-2   7

4   -5

Sol:

Set y1=A1=[-4 -2 4]’,Then r11norm(y1)=6and the first unit vetor is

q1=y1/ norm(y1)=[-2/3 -1/3 2/3]’

To find the second unit vector ,Set

y2=A2-q1q1’A2=[-4 -7 -5]’-[-2/3 -1/3 2/3]’*(-3)=[-6 6 -3]’

and r22=norm(y2)=9,r12=-3

y3=A3-q1q1’A3-q2q2’A3=[1 0 0]’-[-2/3 -1/3 2/3]’(-2/3)- [-2/3 2/3 -1/3]’(-2/3)=[-1 -2/3 2/9]’

and q3=y3/norm(y3)=[-3/11 -2/11 2/33]’

the A= [y1 y2 y3][6 -3

0 9

0 0]

4. Apply modiﬁed Gram–Schmidt orthogonalization to ﬁnd the full QR factorization of the matrices in Exercise 2.

Sol:  a) 2   3

-2   -6

1   0

Set y1=A1=[2 -2 1]’,Then r11norm(y1)=3and the first unit vetor is

q1=y1/ norm(y1)=[2/3 -2/3 1/3]’

To find the second unit vector ,Set

y2=A2-q1q1’A2=[3 -6 0]’-[2/3 -2/3 1/3]’*6=[-1 -2 -2]’

and r22=norm(y2)=3,r12=6

q2=y2/norm(y2)=[-1/3 -2/3 -2/3]’, Adding a third vector A3=[1 0 0] leads to

y3’=A3-q1q1’A3=[1 0 0]’-[2/3 -2/3 1/3]’*1/3=[7/9 2/9 -1/9]’

y3=y3’-q2q2’y3’=[7/9 2/9 -1/9]’-[-1/3 -2/3 -2/3]’(7/27)=[70/81 32/81 7/81]’

A=[y1 y2 y3]*[3 6

0 3

0 0]

b -4   -4

-2   7

4   -5

Solv:

y3’=A3-q1q1’A3=[1 0 0]’-[2/3 1/3 2/3]’*(-2/3)=[5/9 2/9 4/9]’

y3=y3’-q2q2’y3’=[5/9 2/9 4/9]’-[-1/3 2/3 1/3]’*(-2/27)=[42/81 22/81 38/81]’

A=[y1 y2 y3]*[6 -3

0 9

0 0]