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Engineering Computation

Engineering Computation. Lecture 6. Gauss elimination (operation counting). Forward Elimination: DO k = 1 to n–1 DO i = k+1 to n r = A(i,k)/A(k,k) DO j = k+1 to n A(i,j)=A(i,j) – r*A(k,j) ENDDO B(i) = B(i) – r*B(k) ENDDO ENDDO.

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Engineering Computation

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  1. EngineeringComputation Lecture 6 E. T. S. I. Caminos, Canales y Puertos

  2. Gauss elimination (operation counting) Forward Elimination: DO k = 1 to n–1 DO i = k+1 to n r = A(i,k)/A(k,k) DO j = k+1 to n A(i,j)=A(i,j) – r*A(k,j) ENDDO B(i) = B(i) – r*B(k) ENDDO ENDDO E. T. S. I. Caminos, Canales y Puertos

  3. Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Back Substitution: X(n) = B(n)/A(n,n) DO i = n–1 to 1 by –1 SUM = 0 DO j = i+1 to n SUM = SUM + A(i,j)*X(j) ENDDO X(i) = [B(i) – SUM]/A(i,i) ENDDO E. T. S. I. Caminos, Canales y Puertos

  4. Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Forward Elimination Inner loop: Second loop: = (n2 + 2n) – 2(n + 1)k + k2 E. T. S. I. Caminos, Canales y Puertos

  5. Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Forward Elimination (cont'd) Outer loop = E. T. S. I. Caminos, Canales y Puertos

  6. Gauss elimination (operation counting) Operation Counting for Gaussian Elimination Back Substitution Inner Loop: Outer Loop: E. T. S. I. Caminos, Canales y Puertos

  7. Gauss elimination (operation counting) Total flops = Forward Elimination + Back Substitution = n3/3 + O (n2) + n2/2 + O (n)  n3/3 + O (n2) To convert (A,b) to (U,b') requires n3/3, plus terms of order n2 and smaller, flops. To back solve requires: 1 + 2 + 3 + 4 + . . . + n = n (n+1) / 2 flops; Grand Total: the entire effort requires n3/3 + O(n2) flops altogether. E. T. S. I. Caminos, Canales y Puertos

  8. Gauss-Jordan Elimination • Diagonalization by both forward and backward elimination in each column. • Perform elimination both backwards and forwards until: • Operation count for Gauss-Jordan is: • (slower than Gauss elimination) E. T. S. I. Caminos, Canales y Puertos

  9. Gauss-Jordan Elimination Example (two-digit arithmetic): x1 = 0.015 (vs. 0.016, et = 6.3%) x2 = 0.041 (vs. 0.041, et = 0%) x3 = 0.093 (vs. 0.091, et = 2.2%) E. T. S. I. Caminos, Canales y Puertos

  10. Gauss-Jordan Matrix Inversion The solution of: [A]{x} = {b} is: {x} = [A]-1{b} where [A]-1 is the inverse matrix of [A] Consider: [A] [A]-1 = [ I ] 1) Create the augmented matrix: [ A | I ] 2) Apply Gauss-Jordan elimination: ==> [ I | A-1 ] E. T. S. I. Caminos, Canales y Puertos

  11. Gauss-Jordan Matrix Inversion Gauss-Jordan Matrix Inversion (with 2 digit arithmetic): MATRIX INVERSE [A-1] E. T. S. I. Caminos, Canales y Puertos

  12. Gauss-Jordan Matrix Inversion CHECK: [ A ] [ A]-1 = [ I ] [ A]-1 { b } = { x } Gaussian Elimination E. T. S. I. Caminos, Canales y Puertos

  13. LU decomposition • LU decomposition - The LU decomposition is a method that uses the elimination techniques to transform the matrix A in a product of triangular matrices. This is specially useful to solve systems with different vectors b, because the same decomposition of matrix A can be used to evaluate in an efficient form, by forward and backward sustitution, all cases. E. T. S. I. Caminos, Canales y Puertos

  14. Initial system Decomposition Transformed system 1 Substitution Transformed system 2 Backward sustitution Forward sustitution LU decomposition E. T. S. I. Caminos, Canales y Puertos

  15. LU decomposition • LU decomposition is very much related to Gauss method, because the upper triangular matrix is also looked for in the LU decomposition. Thus, only the lower triangular matrix is needed. • Surprisingly, during the Gauss elimination procedure, this matrix L is obtained, but one is not aware of this fact. The factors we use to get zeroes below the main diagonal are the elements of this matrix L. Substract E. T. S. I. Caminos, Canales y Puertos

  16. LU decomposition E. T. S. I. Caminos, Canales y Puertos

  17. LU decomposition (Complexity) Basic Approach Consider [A]{x} = {b} a) Gauss-type "decomposition" of [A] into [L][U] n3/3 flops [A]{x} = {b} becomes [L] [U]{x} = {b}; let [U]{x}{d} b) First solve [L] {d} = {b} for {d} by forward subst. n2/2 flops c) Then solve [U]{x} = {d} for {x} by back substitution n2/2 flops E. T. S. I. Caminos, Canales y Puertos

  18. LU Decompostion: notation [A] = [L] + [U0] [A] = [L0] + [U] [A] = [L0] + [U0] + [D] [A] = [L1] [U] [A] = [L] [U1] E. T. S. I. Caminos, Canales y Puertos

  19. LU decomposition • LU Decomposition Variations • Doolittle [L1][U] General [A] • Crout [L][U1] General [A] • Cholesky [L][L] T Pos. Def. Symmetric [A] • Cholesky works only for Positive Definite Symmetric matrices • Doolittle versus Crout: • • Doolittle just stores Gaussian elimination factors where Crout uses a different series of calculations (see C&C 10.1.4). • • Both decompose [A] into [L] and [U] in n3/3 FLOPS • • Different location of diagonal of 1's • • Crout uses each element of [A] only once so the same array can be used for [A] and [L\U] saving computer memory! E. T. S. I. Caminos, Canales y Puertos

  20. LU decomposition Matrix Inversion Definition of a matrix inverse: [A] [A]-1 = [ I ] ==> [A] {x} = {b} [A]-1 {b} = {x} First Rule:Don’t do it. (numerically unstable calculation) E. T. S. I. Caminos, Canales y Puertos

  21. LU decomposition • Matrix Inversion • If you really must -- • 1) Gaussian elimination: [A | I ] –> [U | B'] ==> A-1 • 2) Gauss-Jordan:[A | I ] ==> [I | A-1 ] • Inversion will take • n3 + O(n2) • flops if one is careful about where zeros are (taking advantage of the sparseness of the matrix) • Naive applications (without optimization) take 4n3/3 + O(n2) flops. For example, LU decomposition requires n3/3 + O(n2) flops. Back solving twice with n unit vectors ei: • 2 n (n2/2) = n3 flops. • Altogether: n3/3 + n3 = 4n3/3 + O(n2) flops E. T. S. I. Caminos, Canales y Puertos

  22. FLOP Counts for Linear Algebraic Equations Summary FLOP Counts for Linear Algebraic Equations, [A]{x} = {b} Gaussian Elimination (1 r.h.s) n3/3 + O (n2) Gauss-Jordan (1 r.h.s) n3/2 + O (n2) LU decomposition n3/3 + O (n2) Each extra LU right-hand-side n2 Cholesky decomposition (symmetric A) n3/6 + O (n2) Inversion (naive Gauss-Jordan) 4n3/3 +O (n2) Inversion (optimal Gauss-Jordan) n3 + O (n2) Solution by Cramer's Rule n! E. T. S. I. Caminos, Canales y Puertos

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