Numerical Methods I: Numerical linear algebra
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1 1/3 Numerical Methods I: Numerical linear algebra Georg Stadler Courant Institute, NYU September 1, 017
2 /3 We study the solution of linear systems of the form Ax = b with A R n n, x, b R n Weassumethatthissystemhasa unique solution, ie, A is invertible is needed in many applications Often, we have to solve I large systems (can be up to millions of unknowns, and more) I as fast as possible, and I accurately and reliably There exist explicit formulas for solving linear systems but they are extremely expensive (eg, Kramer s rule requires computing determinants)
3 Triangular systems (forward substitution): l x 1 b 1 l 1 l = l n1 l nn x n b n 3/3
4 4/3 Triangular systems, implementation: l x 1 b 1 l 1 l = l n1 l nn x n b n
5 5/3 Triangular systems: Forward and backward substitution, requires n(n + 1) n(n 1) multiplications/divisions, additions Overall: n floating point operations (flops) We count flops to estimate the computational time/e ort Besides floating point operations, computer memory access has a significant influence on the e ciency of numerical methods (see experiments in homework #)
6 Gaussian elimination LU factorization 6/3 3 3 a 11 a 1n x a n1 a nn x n 7 5 = 6 4b 1 Gaussian elimination: new row = row i - l i1 row a 11 a 1n x 1 b 1 0 a 0 a 0 n x b 0 = b a 0 n a 0 nn x n b 0 n New system matrix/rhs is: A () = L 1 A, b () = L 1 b b n 3 7 5
7 Gaussian elimination LU factorization 6/3 3 3 a 11 a 1n x a n1 a nn x n 7 5 = 6 4b 1 Gaussian elimination: new row = row i - l i1 row a 11 a 1n x 1 b 1 0 a 0 a 0 n x b 0 0 a a 00 3n = b a 00 n3 a 00 nn x n b 00 n New system matrix/rhs is: A (3) = L L 1 A, b (3) = L L 1 b b n 3 7 5
8 Gaussian elimination LU factorization 7/3 We obtain: A (n) = L n 1 L 1 A, b (n) = L n 1 L 1 b, with the Frobenius matrices 3 1 L k = 1 l k+1,k l n,k 1 Note that L 1 k for the l j,i s are also Frobenius matrices, but with di erent sign
9 Gaussian elimination LU factorization We obtain: A (n) = L n 1 L 1 A, b (n) = L n 1 L 1 b, with the Frobenius matrices 3 1 L k = 1 l k+1,k l n,k 1 Note that L 1 k for the l j,i s are also Frobenius matrices, but with di erent sign 8/3
10 3 a 11 a 1 a 1n a 1 a a n a n1 a n a nn u 11 u 1 u 1n l u u n = l n1 l n u nn u 11 = a 11 u 1 = a 1 l 1 u 11 = a 1 l 1 u 1 + u = a 9/3
11 Gaussian elimination LU factorization 10 / 3 Algorithm for solving linear system Ax = b (assuming diagonal elements do not vanish): 1 Compute triangular factorization A = LU Solve Lz = b (forward substitution) 3 Solve U x = z (backward substitution)
12 Gaussian elimination LU factorization Algorithm for solving linear system Ax = b (assuming diagonal elements do not vanish): 1 Compute triangular factorization A = LU Solve Lz = b (forward substitution) 3 Solve U x = z (backward substitution) Notes: I Main cost is LU factorization I Factorization can be reused for di erent right hand sides b 10 / 3
13 11 / 3 LU with pivoting If diagonal pivoting element is zero (or very small), one has to exchange rows and/or columns otherwise the LU factorization fails Basic idea: Choose largest element (in absolute value) in the row that is eliminated as pivot
14 1 / 3 LU with pivoting Example with a 3 digit computer:
15 LU with pivoting Pivoting can be expressed by permutation matrices P,resultingin the LU decomposition (the permutation also a ects L and U) Theorem: For every invertible matrix A, there exists a permutation matrix P such that P A = LU is possible The permutation can be chosen such that all entries in L are apple 1 13 / 3
16 LU with pivoting 14 / 3
17 Choleski factorization 15 / 3 Amatrixissymmetric positive definite (spd), ifa = A T and for all x R n, x 6= 0,theinnerproducthAx, xi > 0 For spd matrices, we can compute the factorization: A = LDL T, where L is a lower triangular matrix with 1 s on the diagonal, and D is a positive diagonal matrix The Choleski factorization is obtained by multiplying the square root of D (which exists!) with L: A = L L T Choleski factorization requires n3 6 roots multiplications and n square
18 Kinds of linear systems 16 / 3 Solvers such as MATLAB s \ take advantage of matrix properties: I Dense matrix storage: Only entries are stored as 1D array (column or row wise) I Sparse matrix storage: Most a ij =0: only store nonzero entries; stores indices and value; occur in many applications
19 Kinds of linear systems Solvers such as MATLAB s \ take advantage of matrix properties: 17 / 3 I Fast algorithms for special matrices: for computing Ax, FFT, FMM, I Sparse: Most a ij =0: avoid fill-in in factorizations I Structured/unstructured: isthesparsitypatterneasyto describe without storing it explicitly?
20 inds of linear systems and solvers 18 / 3 Symmetry, positivity I Special factorizations for (skew) symmetric matrices I Special factorizations for positive definite matrices (Choleski) I Diagonally dominant matrices don t need pivoting MATLAB s \ (ie, UMFPACK) chooses the optimal algorithms after studying properties of the matrix (details in the backslash book: Tim Davis: Direct methods for sparse linear systems, SIAM, 006)
21 Kinds of linear systems and solvers UMFPACK s decision tree for dense matrices 19 / 3
22 Kinds of linear systems and solvers UMFPACK s decision tree for dense matrices 19 / 3
23 Kinds of linear systems and solvers UMFPACK s decision tree for sparse matrices 0 / 3
24 Kinds of linear systems and solvers UMFPACK s decision tree for sparse matrices 0 / 3
25 inds of linear systems and solvers 1 / 3 Factorization-based/direct solvers (dense/sparse LU, Choleski) require the matrix - to fit into memory, - to be explicitly available (sometimes only a function that applies the matrix to a vector is available) and to fit in memory, + but compute exact (besides rounding error) solution Iterative solvers - find an "-approximation of the solution, + able to solve very large problems, + often only require a function that computes Ax for given x ± might be faster or slower than a factorization-based method
26 Kinds of linear systems and solvers MATLAB demo / 3 I What are the di erent storage formats (sparse/dense)? Is it always better to use one of them? I How long does it take to solve sparse/dense systems? I What is fill in and how to avoid it?
27 Kinds of linear systems and solvers MATLAB demo 3 / 3 Sparse/sense storage: A=rand(,); B=sparse(A); whos Fill-in: A=bucky + 4*speye(60); r = symrcm(a); spy(a); spy(a(r,r)); spy(chol(a)); spy(chol(a(r,r))); Which sparse solver? spparms( spumoni,1); A=gallery( poisson,8); b=randn(64,1); A\b;
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