Numerical Solution of Differential Algebraic Equations

... have been written as part of a Ph.D course on the numerical solution of Differential Algebraic Equations The course was held at IMM in the fall of 1998 The authors of the different chapters have ... PREFACE CHAPTER Introduction The (modern) theory of numerical solution of ordinary differential equations (ODEs) has been developed since the early part of this cen...

Ngày tải lên: 21/12/2016, 10:52

... 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,...

Ngày tải lên: 15/12/2013, 04:15

... 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...

Ngày tải lên: 15/12/2013, 04:15

... 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 +...

Ngày tải lên: 15/12/2013, 04:15

... =  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...

Ngày tải lên: 24/12/2013, 12:16

... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 equation...

Ngày tải lên: 24/12/2013, 12:16

... 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: Pre...

Ngày tải lên: 24/12/2013, 12:16

... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 L...

Ngày tải lên: 24/12/2013, 12:16

... 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 Softw...

Ngày tải lên: 21/01/2014, 18:20

... 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...

Ngày tải lên: 21/01/2014, 18:20