1. 數值方法, Applied Mathematics NDHU 1 Linear system

2. 數值方法, Applied Mathematics NDHU 2 Linear system

3. 數值方法, Applied Mathematics NDHU 3 m=n If A is invertible

4. 數值方法, Applied Mathematics NDHU 4 inv >> A=rand(5,5);b=rand(5,1); >> x=inv(A)*b x = -2.2355 9.2038 -7.0138 -2.8158 13.3273

5. 數值方法, Applied Mathematics NDHU 5 Naive Gaussian elimination Direct method Analytic approach Naive Gaussian elimination Iterative method Jacobi method Gauss-Seidel method SOR method

6. 數值方法, Applied Mathematics NDHU 6 Forward elimination r1*(-2)+r2 r1*(-1/2)+r3 r1+r3 r2*(-3)+r3 r2*(1/2)+r4 r3*(-2)+r4

7. 數值方法, Applied Mathematics NDHU 7 Triangular linear system

8. 數值方法, Applied Mathematics NDHU 8 Backward substitution

9. 數值方法, Applied Mathematics NDHU 9 Procedure : Naive forward elimination Input a square matrix A,b Set n to the row number of A B=[A b] for i=1 to n for j = i+1 to n Set c to the ratio B(j,i) / B(i,i) Set v to the product of c and the ith row of B Subtract v from the jth row of B Return B

10. 數值方法, Applied Mathematics NDHU 10 B=Gauss_eli(A,b) for i=1:n-1 for j=i+1:n n=length(b); B=[A b]; exit c=B(j,i)/B(i,i) B(j,:)=B(j,:)-c*B(i,:)

11. 數值方法, Applied Mathematics NDHU 11 Procedure : Backward substitution input an upper triangle matrix U and a column vector b set n to the row number of U for i = n to 1 set v to b(i) for j = n to i+1 subtract the product of U(i,j) and x(j) from v set x(i) to v/U(i,i)

12. 數值方法, Applied Mathematics NDHU 12 x=backward_sub(U,b) for i=n:-1:1 for j=n:-1:i+1 n=length(b); exit v=v-U(i,j)*x(j) v=b(i) x(i)=v/U(i,i)

13. 數值方法, Applied Mathematics NDHU 13 Exercise Implement naive forward elimination and backward substitution for solving a linear system Give two examples to test your matlab codes

14. 數值方法, Applied Mathematics NDHU 14 m > n The size of unknowns in a linear system is less than the number of linear constraints.

15. 數值方法, Applied Mathematics NDHU 15

16. 數值方法, Applied Mathematics NDHU 16 A = 0.9501 0.6154 0.0579 0.2311 0.7919 0.3529 0.6068 0.9218 0.8132 0.4860 0.7382 0.0099 0.8913 0.1763 0.1389 0.7621 0.4057 0.2028 0.4565 0.9355 0.1987 0.0185 0.9169 0.6038 0.8214 0.4103 0.2722 0.4447 0.8936 0.1988 b = 0.7273 -0.7687 0.1832 -0.4946 1.5690 0.9155 -0.7593 -1.1930 1.0945 -0.6991

17. 數值方法, Applied Mathematics NDHU 17 Least square errors

18. 數值方法, Applied Mathematics NDHU 18 Minimization

19. 數值方法, Applied Mathematics NDHU 19 Derivation

20. 數值方法, Applied Mathematics NDHU 20

21. 數值方法, Applied Mathematics NDHU 21

22. 數值方法, Applied Mathematics NDHU 22

23. 數值方法, Applied Mathematics NDHU 23

24. 數值方法, Applied Mathematics NDHU 24 Solving linear systems with m>n Input paired data Form matrix A and vector b Set x1 to pinv(A)*b Set x2 to

25. 數值方法, Applied Mathematics NDHU 25 >> A=rand(30,2);b=rand(30,1); >> x1=pinv(A)*b; >> x2=inv(A'*A)*(A'*b); >> sum(abs(x1-x2)) ans = 1.0547e-015