Numerical Analysis EE NCKU TienHao Chang Darby Chang
Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang) 1
In the previous slide n 2
In this slide n Special matrices – strictly diagonally dominant matrix – symmetric positive definite matrix • Cholesky decomposition – tridiagonal matrix n Iterative techniques – Jacobi, Gauss-Seidel and SOR methods – conjugate gradient method n n Nonlinear systems of equations (Exercise 3) 3
3. 7 Special matrices 4
Special matrices n Linear systems – which arise in practice and/or in numerical methods – the coefficient matrices often have special properties or structure n Strictly diagonally dominant matrix n Symmetric positive definite matrix n Tridiagonal matrix 5
Strictly diagonally dominant 6
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Symmetric positive definite 8
Symmetric positive definite Theorems for verification 9
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Symmetric positive definite Relations to n Eigenvalues n Leading principal sub-matrix 11
Cholesky decomposition n 12
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Tridiagonal n 15
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Any Questions? 3. 7 Special matrices 17
Before entering 3. 8 n So far, we have learnt three methods algorithms in Chapter 3 – Gaussian elimination – LU decomposition – direct factorization n n question Are they algorithms? What’s the differences to those algorithms further question in Chapter 2? – they report exact solutions rather than answer approximate solutions 18
Before entering 3. 8 n So far, we have learnt three methods algorithms in Chapter 3 – Gaussian elimination – LU decomposition – direct factorization n n Are they algorithms? What’s the differences to those algorithms further question in Chapter 2? – they report exact solutions rather than answer approximate solutions 19
Before entering 3. 8 n So far, we have learnt three methods algorithms in Chapter 3 – Gaussian elimination – LU decomposition – direct factorization n n Are they algorithms? What’s the differences to those algorithms in Chapter 2? – they report exact solutions rather than answer approximate solutions 20
Before entering 3. 8 n So far, we have learnt three methods algorithms in Chapter 3 – Gaussian elimination – LU decomposition – direct factorization n n Are they algorithms? What’s the differences to those algorithms in Chapter 2? – they report exact solutions rather than approximate solutions 21
3. 8 Iterative techniques for linear systems 22
Iterative techniques n 23
Iterative techniques Basic idea 24
Iteration matrix Immediate questions n 25
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n (in section 2. 3 with proof) Recall that http: //www. dianadepasquale. com/Thinking. Monkey. jpg 29
Recall that http: //www. dianadepasquale. com/Thinking. Monkey. jpg 30
Iteration matrix For these questions n question hint answer 31
Iteration matrix For these questions n hint answer 32
Iteration matrix For these questions n answer 33
Iteration matrix For these questions n 34
Splitting methods 35
Splitting methods n 36
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Gauss-Seidel method 39
Gauss-Seidel method Iteration matrix 40
The SOR method (successive overrelaxatoin) 41
Any Questions? Iterative techniques for linear systems 42
3. 9 Conjugate gradient method 43
Conjugate gradient method n n Not all iterative methods are based on the splitting concept The minimization of an associated quadratic functional 44
Conjugate gradient method Quadratic functional 45
http: //fuzzy. cs. uni-magdeburg. de/~borgelt/doc/somd/parabola. gif 46
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Minimizing quadratic functional 48
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http: //www. mathworks. com/cmsimages/op_main_wl_3250. jpg Global optimization problem 50
Any Questions? Conjugate gradient method 51
3. 10 Nonlinear systems of equations 52
Nonlinear systems of equations 53
Generalization of root-finding 54
Generalization Newton’s method 55
Generalization of Newton’s method Jacobian matrix 56
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http: //www. math. ucdavis. edu/~tuffley/sammy/Lin. Alg. DEs 1. jpg A lots of equations bypassed… 58
And this is a friendly textbook : ) 59
Any Questions? Nonlinear systems of equations 60
Exercise 3 2011/5/2 2: 00 pm Email to darby@ee. ncku. edu. tw or hand over in class. Note that the fourth problem is a programming work. 61
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Implement LU decomposition 65
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