LU Factorization. LU Decomposition. LU Decomposition. LU Decomposition: Motivation A = LU
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1 LU Factorization To further improve the efficiency of solving linear systems Factorizations of matrix A : LU and QR LU Factorization Methods: Using basic Gaussian Elimination (GE) Factorization of Tridiagonal Matrix Using Gaussian Elimination with pivoting Direct LU Factorization Factorizing Symmetrix Matrices (Cholesky Decomposition) Applications Analysis LU Decomposition A matrix A can be decomposed into a lower triangular matrix L and upper triangular matrix U so that A = LU LU decomposition is performed once; can be used to solve multiple right hand sides. Similar to Gaussian elimination, care must be taken to avoid roundoff errors (partial or full pivoting) Special Cases: Banded matrices, Symmetric matrices. ITCS 4133/5133: Intro. to Numerical Methods 1 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 2 LU/QR Factorization LU Decomposition: Motivation LU Decomposition Forward pass of Gaussian elimination results in the upper triangular matrix 1 u 12 u 13 u 1n 0 1 u 23 u 2n U = u 3n We can determine a matrix L such that LU = A, and ITCS 4133/5133: Intro. to Numerical Methods 3 LU/QR Factorization l l 21 l L = l 31 l 32 l l n1 l n2 l n3 1 ITCS 4133/5133: Intro. to Numerical Methods 4 LU/QR Factorization
2 LU Decomposition via Basic Gaussian Elimination LU Decomposition via Basic Gaussian Elimination: Algorithm ITCS 4133/5133: Intro. to Numerical Methods 5 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 6 LU/QR Factorization Banded Matrices LU Decomposition of Tridiagonal Matrix Matrices that have non-zero elements close to the main diagonal Example: matrix with a bandwidth of 3 (or half band of 1) a 11 a a 21 a 22 a a 32 a 33 a a 43 a 44 a a 54 a 55 Efficiency: Reduced pivoting needed, as elements below bands are zero. Iterative techniques are generally more efficient for sparse matrices, such as banded systems. ITCS 4133/5133: Intro. to Numerical Methods 7 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 8 LU/QR Factorization
3 LU Decomposition of Tridiagonal Matrix: Algorithm LU Decomposition via GE with Pivoting ITCS 4133/5133: Intro. to Numerical Methods 9 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 10 LU/QR Factorization LU Decomposition via GE with Pivoting: Algorithm Direct LU Decomposition l u 12 u 13 u 1n a 11 a 12 a 13 a 1n l 21 l u 23 u 2n a 21 a 22 a 23 a 2n l 31 l 32 l u 3n = a 31 a 32 a 33 a 2n l n1 l n2 l n a n1 a n2 a n3 a nn The above equation shows the coefficient matrix A can be decomposed into L and U; L and U can be determined by the computing the product and equating like terms. ITCS 4133/5133: Intro. to Numerical Methods 11 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 12 LU/QR Factorization
4 Direct LU Decomposition: Example Determining L and U l i1 u 1j l ij u ji l nn = a i1, i = 1, 2,..., n = a 1j, j = 2, 3,..., n l 11 j 1 = a ij l ik u kj, k=1 for j = 2, 3,..., n 1 and i = j, j + 1,..., n = a ji j 1 l k=1 jku ki, l jj for j = 2, 3,..., n 1 and i = j + 1, j + 2,..., n n 1 = a nn l nk u kn k=1 ITCS 4133/5133: Intro. to Numerical Methods 13 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 14 LU/QR Factorization Direct LU Decomposition: Algorithm Symmetric Matrices Symmetric square matrices - common in engineering, for example stiffness matrix (stiffness properties of structures). For a symmetric matrix A A = A T where A T is the transpose of A. Hence For a symmetric matrix A A = LL T ITCS 4133/5133: Intro. to Numerical Methods 15 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 16 LU/QR Factorization
5 Symmetric Matrices: LU Decomposition LU (Cholesky) Decomposition of Symmetric Matrices We can use Cholesky Decomposition to compute the LU decomposition of A l 11 = a 11 l ii = l ij i 1 aii lik 2, for i = 2, 3,..., n k=1 = a ij j 1 k=1 l ikl jk l jj, for j = 2, 3,..., i 1, and j < 1 ITCS 4133/5133: Intro. to Numerical Methods 17 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 18 LU/QR Factorization LU (Cholesky) Decomposition: Algorithm Applications of LU Decomposition:Solving Linear systems where e = [U] X [A] X = [L][U][ X] = C = [L] e = C Solve for e using L (forward substitution) Solve for X using U (back substitution) ITCS 4133/5133: Intro. to Numerical Methods 19 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 20 LU/QR Factorization
6 Solving Linear Systems Solving Linear Systems: Example Forward Substitution Solve for e, e 1 = C 1 l 11 = C i n e i Back Substitution Solve for X l ii j=1 l ije j, for i = 2, 3,... n X n X i = e n n = e i u ij X j, for i = n 1, n 2,... 1 j=i+1 ITCS 4133/5133: Intro. to Numerical Methods 21 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 22 LU/QR Factorization Solving Linear Systems: Algorithm Application 2: Tridiagonal Systems ITCS 4133/5133: Intro. to Numerical Methods 23 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 24 LU/QR Factorization
7 Application 3: Matrix Inverse Analysis: LU Decomposition Why does LU decomposition work? Consider the first elimination step: m a 11 a 12 a 13 a 11 a 12 a 13 a 21 a 22 a 23 = 0 a 22 a 23 m a 31 a 32 a 33 0 a 32 a 33 Define M 1 = m , M 2 = m m 32 1 ITCS 4133/5133: Intro. to Numerical Methods 25 LU/QR Factorization Hence M 1 1 = m , M 1 2 = , m m 32 1 ITCS 4133/5133: Intro. to Numerical Methods 26 LU/QR Factorization Analysis: LU Decomposition Gradual tranformation of I towards LU: I.A = A I.M 1 1.M 1.A = A I.M 1 1.M 2 1.M 2.M 1.A = A M 2.M 1.A is U and I.M 1 1.M 1 2 is L, given by m m 31 m 32 1 ITCS 4133/5133: Intro. to Numerical Methods 27 LU/QR Factorization Analysis:LU Decomposition with Pivoting Decomposition results in the factorization of P A, where P is the permutation matrix. [ ] [ a11 ] a 12 a 13 = a 21 a 22 a 23 a 31 a 32 a 33 [ a11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Assume no pivoting in column 1, pivoting in column 2: I.A = A I.M 1 1.M 1.A = A I.M 1 1.P.P.M 1.A = A I.M 1 1.P.M 2 1.M 2.P.M 1.A = A M 2.P.M 1.A is upper triangular, but I.M 1 1.P.M 2 1 is not lower triangular. Premultiply both sides by P : P M 1 1.P.M 2 1.M 2.P.M 1.A = P.A ITCS 4133/5133: Intro. to Numerical Methods 28 LU/QR Factorization ]
8 ITCS 4133/5133: Intro. to Numerical Methods 29 LU/QR Factorization
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![MODULE 7. where A is an m n real (or complex) matrix. 2) Let K(t, s) be a function of two variables which is continuous on the square [0, 1] [0, 1].](/thumbs/95/122529712.jpg "MODULE 7. where A is an m n real (or complex) matrix. 2) Let K(t, s) be a function of two variables which is continuous on the square [0, 1] [0, 1].")
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