solving systems of linear equations using matrix inverse methods

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Ngày tải lên: 20/06/2014, 22:20

... there are matrix elements in A 1 , ,A n . Let us call P (z) the monodromy matrix of (1). Other results of Birkhoff show that for any periodic matrix P of a specific form, there exists an equation of the ... Sp(S)=Sp(Y ), Sp(T ) = Sp(X). The matrix elements of S and T are rational functions of the matrix elements of X and Y which do not depend on the spectra of X and Y . Proof 1. Lemma 3.3 proves the uniqueness ... Y σ(1) ) ···(z −Y σ(N) ). The matrix elements of {Y i } are rational functions of the matrix elements of {X i } which do not depend on the spectra of {X i }. Proof. The existence and rationality...

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Ngày tải lên: 05/09/2013, 15:28

... columns of the matrix inverse of A (Đ2.1 and Đ2.3). ã Calculation of the determinant of a square matrix A (Đ2.3). If M<N,orifM=Nbut the equations are degenerate, then there are effectively fewer equations ... simplification is that the matrix A is held constant, while the right-hand sides, the bs, are changed (Đ2.1Đ2.10). ã Calculation of the matrix A −1 which is the matrix inverse of a square matrix A, i.e., ... equations as unknowns, and there is a good chance of solving for a unique solution set of x j ’s. Analytically, there can fail to be a unique solution if one or more of the M equations is a linear...

Ngày tải lên: 15/12/2013, 04:15

... C. 1971, Linear Algebra ,vol.IIof Handbook for Automatic Com- putation (New York: Springer-Verlag). Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New ... inverting a matrix, Gauss-Jordan elimination is about as efficient as any other method. For solving sets of linear equations, Gauss-Jordan elimination produces both the solution of the equations ... =1,2,3,4) of the vector solution of the jth right-hand side (j =1,2,3), the one whose coefficients are b ij ,i =1,2,3,4;and that the matrix of unknown coefficients y ij is the inverse matrix of a ij .Inother words,...

Ngày tải lên: 15/12/2013, 04:15

... is not used for typical systems of linear equations. However, we will meet special cases where QR is the method of choice. 100 Chapter 2. Solution of Linear Algebraic Equations Sample page from ... involve solvinga successionof linear systems each of which differs only slightly from its predecessor. Instead of doing O(N 3 ) operations each time to solve the equations from scratch, one can often ... solve linear systems. In many applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98 Chapter 2. Solution of Linear Algebraic Equations Sample...

Ngày tải lên: 15/12/2013, 04:15

... to a factor 1.5 by not computing the inverse matrix as part of the Gauss-Jordan scheme.) For computing the inverse matrix (which we can view as the case of M = N right-hand sides, namely the ... York: McGraw-Hill), Program B-2, p. 298. Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley). Ralston, A., and Rabinowitz, P. 1978, A ... 42 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C)...

Ngày tải lên: 24/12/2013, 12:16

... Johnson, C.R. 1985, Matrix Analysis (Cambridge: Cambridge University Press). 2.4 Tridiagonal and Band Diagonal Systems of Equations The special case of a system of linear equations that is tridiagonal, ... columns of B instead of with the unit vectors that would give A’s inverse. This saves a whole matrix multiplication, and is also more accurate. Determinant of a Matrix The determinant of an LU ... modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors...

Ngày tải lên: 24/12/2013, 12:16

... Johnson, C.R. 1985, Matrix Analysis (Cambridge: Cambridge University Press). 2.4 Tridiagonal and Band Diagonal Systems of Equations The special case of a system of linear equations that is tridiagonal, ... Prentice-Hall), Chapters 9, 16, and 18. Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley). Stoer, J., and Bulirsch, R. 1980, Introduction ... limitations of bandec, and the above routine does take advantage of the opportunity. In general, when TINY is returned as a diagonal element of U, then the original matrix (perhaps as modified by roundoff...

Ngày tải lên: 24/12/2013, 12:16

... submatrices. Imagine doing the inversionof a very large matrix, of order N =2 m , recursively by partitions in half. At each step, halving the order doubles the number of inverse operations. But this means ... 104 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) ... it is that factor at each hierarchical level of the recursion. In total it reduces the process of matrix multiplication to order N log 2 7 instead of N 3 . What about all the extra additions in...

Ngày tải lên: 24/12/2013, 12:16

... n] of the linear set of equations A · X = B.Thematrix a[1 n][1 n] , and the vectors b[1 n] and x[1 n] are input, as is the dimension n . Also input is alud[1 n][1 n] ,theLU decomposition of a as ... decomposition of A is itself not exact. A different analytical approach starts with some matrix B 0 that is assumed to be an approximate inverse of the matrix A,sothatB 0 ·Ais approximately the identity matrix ... 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 computer’s floating-point...

Ngày tải lên: 24/12/2013, 12:16

... Sparse Linear Systems A system of linear equations is called sparse if only a relatively small number of its matrix elements a ij are nonzero. It is wasteful to use general methods of linear ... making the same permutation of the columns of U,elementsofW,andcolumnsofV(or rows of V T ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to be ... throwing away one linear combination of the set of equations that we are trying to solve. The resolution of the paradox is that we are throwing away precisely a combination of equations that is...

Ngày tải lên: 24/12/2013, 12:16

... Sparse Linear Systems A system of linear equations is called sparse if only a relatively small number of its matrix elements a ij are nonzero. It is wasteful to use general methods of linear ... applications.) ã Each of the rst N locations of ija stores the index of the array sa that contains the first off-diagonal element of the corresponding row of the matrix. (If there are no off-diagonal elements ... 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 3 for the general linear...

Ngày tải lên: 24/12/2013, 12:16

... Interval-based Solving of Hybrid Constraint Systems. 3rd IFAC Conference on Analysis and Design of Hybrid Systems (ADHS09) (to appear) 10. Kăuhn, W.: Rigorously Computed Orbits of Dynamical Systems ... syst`emes hybrides, equations differentielles ordinaires, anal- yse par intervalles. References 1. Hansen, E. and Sengupta, S.: Bounding solutions of systems of equations using interval analysis. ... BIT 21, 203–211 (1981) 2. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, (1990) 3. Goldsztejn, A.: A Comparison of the Hansen-Sengupta and Frommer-Lang-Schnurr Existence...

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