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Tài liệu Solution of Linear Algebraic Equations part 1 docx

Tài liệu Solution of Linear Algebraic Equations part 1 docx

... inversion” (§2 .11 ) 36 Chapter Solution of Linear Algebraic Equations Coleman, T.F., and Van Loan, C 19 88, Handbook for Matrix Computations (Philadelphia: S.I.A.M.) Forsythe, G.E., and Moler, C.B 19 67, ... right-hand sides, the b’s, are changed (§2 .1 §2 .10 ) • Calculation of the matrix A 1 which is the matrix inverse of a square matrix A, i.e., A · A 1 = A 1 · A = 1, where is the identity matrix (all ... can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution xp added to any linear combination of (typically) N...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt

Tài liệu Solution of Linear Algebraic Equations part 2 ppt

... b11  b21  b31 b41   b 12  b 22  b 32 b 42    x 12  x 22  x 32 x 42  x13  x23  x33 x43  b13  b23  b33 b43  0 0 y11  y21 y31 y41 0 0 y 12 y 22 y 32 y 42 y13 y23 y33 y43  y14 y24  y34 ... 37 2. 1 Gauss-Jordan Elimination Elimination on Column-Augmented Matrices  Consider the linear matrix equation      a 12 a 22 a 32 a 42 a13 a23 a33 a43 a14 x11 a24   x21  · a34 ... equation that looks like this (in the case of a single right-hand side vector):       x1 b1 a11 a 12 a13 a14  a 22 a23 a24   x2   b2  (2. 2.1) ·  =    0 a33 a34 x3 b3 0 a44 x4...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt

Tài liệu Solution of Linear Algebraic Equations part 11 ppt

... multiplication of two × matrices, a11 a21 a12 a22 · b11 b21 b12 b22 = c11 c21 c12 c22 (2 .11. 1) Eight, right? Here they are written explicitly: c11 = a11 × b11 + a12 × b21 c12 = a11 × b12 + a12 ... set of formulas was, in fact, discovered by Strassen [1] The formulas are: Q1 ≡ (a11 + a22 ) × (b11 + b22 ) Q2 ≡ (a21 + a22 ) × b11 Q3 ≡ a11 × (b12 − b22 ) Q4 ≡ a22 × (−b11 + b21 ) Q5 ≡ (a11 + ... column of A below the first element Similarly Q2 zeroes all elements in the second column below the second element, and so on up to Qn−1 Thus 100 Chapter Solution of Linear Algebraic Equations...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf

Tài liệu Solution of Linear Algebraic Equations part 3 pdf

... =  21 a31 a41 a12 a22 a32 a42 a 13 a 23 a 33 a 43 a14 a24  a34 a44 (2 .3. 2) We can use a decomposition such as (2 .3. 1) to solve the linear set A · x = (L · U) · x = L · (U · x) = b (2 .3. 3) by first ... For the case of a × matrix A, for example, equation (2 .3. 1) would look like this:       α11  α21 31 α41 α22 32 α42 0 33 α 43 β11   · 0 α44 β12 β22 0 β 13 β 23 33 β14 β24  34 β44 a11 ... Chapter Solution of Linear Algebraic Equations Backsubstitution But how we solve for the x’s? The last x (x4 in this example) is already isolated, namely x4 = b4 /a44 (2.2.2) x3 = [b − x4 a34 ] a33...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx

Tài liệu Solution of Linear Algebraic Equations part 4 docx

... brief, Crout’s method fills in the combined matrix of α’s and β’s,  β11  α21  α31 41 β12 β22 α32 42 β13 β23 β33 43  β 14 β 24   β 34 44 (2.3. 14) by columns from left to right, and within each ... in the reduction free_vector(vv,1,n); 48 Chapter Solution of Linear Algebraic Equations To summarize, this is the preferred way to solve the linear set of equations A · x = b: float **a,*b,d; ... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx

Tài liệu Solution of Linear Algebraic Equations part 5 docx

... improved solution x 2 .5 Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your ... number of right-hand sides can be solved in turn by repeated calls to banbks, the backsubstitution routine whose analog in §2.3 is lubksb 55 2 .5 Iterative Improvement of a Solution to Linear Equations ... tridiagonal linear set given by equation (2.4.1) a[1 n], b[1 n], c[1 n], and r[1 n] are input vectors and are not modified { unsigned long j; float bet,*gam; 52 Chapter Solution of Linear Algebraic Equations...
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Tài liệu Solution of Linear Algebraic Equations part 12 pdf

Tài liệu Solution of Linear Algebraic Equations part 12 pdf

... to trade@cup.cam.ac.uk (outside North America) c22 = Q1 + Q3 − Q2 + Q6 104 Chapter Solution of Linear Algebraic Equations CITED REFERENCES AND FURTHER READING: Strassen, V 1969, Numerische Mathematik, ... that the matrices a11 a21 a12 a22 and c11 c21 c12 c22 (2.11.5) are inverses of each other Then the c’s can be obtained from the a’s by the following operations (compare equations 2.7.22 and 2.7.25): ... submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively by partitions in half At each step, halving the order doubles the number of inverse operations But this means...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx

Tài liệu Solution of Linear Algebraic Equations part 6 pptx

... inverse of the matrix A, so that B0 · A is approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix ... 56 Chapter Solution of Linear Algebraic Equations But (2.5.2) can also be solved, trivially, for δb Substituting this ... Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall), §5.5 .6, p 183 Forsythe, G.E., and Moler, C.B 1 967 , Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapter...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx

Tài liệu Solution of Linear Algebraic Equations part 7 docx

... of the elements wj From (2.6.1) it now follows immediately that the inverse of A is 62 Chapter Solution of Linear Algebraic Equations If we want to single out one particular member of this solution- set ... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... solutions of A⋅x = d solutions of A ⋅ x = c′ SVD solution of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular matrix A maps a vector space into one of the...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx

Tài liệu Solution of Linear Algebraic Equations part 8 docx

... unlikely event that this causes loss of precision in the second of these equations, you can make a different choice 76 Chapter Solution of Linear Algebraic Equations Here A is, as usual, an N ... http://www.nr.com or call 1 -80 0 -87 2-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) Inversion by Partitioning 78 Chapter Solution of Linear Algebraic Equations simpler ... because it corresponds to successive minimizations of the function 86 Chapter Solution of Linear Algebraic Equations and modifies the definitions of αk , βk , pk , and pk in equation (2.7.32): rk...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx

Tài liệu Solution of Linear Algebraic Equations part 9 docx

... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 198 8- 199 2 by Cambridge University Press.Programs Copyright (C) 198 8- 199 2 by Numerical Recipes Software Permission ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 198 8- 199 2 by Cambridge University Press.Programs Copyright (C) 198 8- 199 2 by Numerical Recipes Software Permission ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 198 8- 199 2 by Cambridge University Press.Programs Copyright (C) 198 8- 199 2 by Numerical Recipes Software Permission...
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Tài liệu Solution of Linear Algebraic Equations part 10 docx

Tài liệu Solution of Linear Algebraic Equations part 10 docx

... case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N for the general linear ... diagonal) part of A, preserving the input upper triangular values of A Only one extra vector of length N is needed to store the diagonal part of L The operations count is N /6 executions of the ... forms] Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New...
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Tài liệu Báo cáo

Tài liệu Báo cáo " Fully parallel methods for a class of linear partial differential-algebraic equations " pptx

... Galo, I Albarreal, M.C Calzada, J.L Cruz, E Fernandez-Cara, M Mari´n, Stability and Convergence of a Parallel ´ Fractional Step Method for the Solution of Linear Parabolic Problems, Applied Mathematics ... Marszalek, Z Trzaska, A Boundary-value Problem for Linear PDAEs, Int.J.Appl.Math.Comput.Sci., Vol 12, No (2002) 487 [3] T Lu, P Neittaanmaki, X.C Tai, A Parallel Splitting-up Method for Partial ... Adams, New Parallel Method by Domain Partitioning, SIAM J Sci Comput Vol 20, No (1999) 2261 [9] C.Zhang, Hong Lan, Y.E.Yang, B.D Estrade, Parallel SOR Iterative Algorithms and Performance Evaluation...
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A FUNCTIONAL-ANALYTIC METHOD FOR THE STUDY OF DIFFERENCE EQUATIONS EUGENIA N. PETROPOULOU AND potx

A FUNCTIONAL-ANALYTIC METHOD FOR THE STUDY OF DIFFERENCE EQUATIONS EUGENIA N. PETROPOULOU AND potx

... to study partial difference equations of three and four variables The aim of this paper is to present the generalization of this functional-analytic method for the study of linear and nonlinear ... to apply Theorem 1.1 to the preceding operator equation and obtain information for the initial linear difference equation under consideration In the case of nonlinear equations, we some manipulation ... generating analytic functions, Laurent or z-transforms, numerical schemes, boundary value problems of partial differential equations, problems of quantum mechanics, and problems of population dynamics...
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