solving systems of linear equations by matrix methods

... classes of nonlinear matrix equations (see [8-21]) In this study, we consider the following problem: Find (X1, X2, , Xm) Î (P(n))m solution to the following system of nonlinear matrix equations: ... Two kinds of nonlinear matrix equations and their corresponding matrix sequences Linear Multilinear Algebra 52, 1–15 (2004) doi:10.1080/0308108031000112606 Zhan, X, Xie, J: On the matrix equation ... method for solving a class of nonlinear matrix equation Appl Math Comput 216, 1831–1836 (2010) doi:10.1016/j.amc.2009.12.022 Liu, X, Gao, H: On the positive definite solutions of the matrix equations...

Ngày tải lên: 20/06/2014, 22:20

... Annals of Mathematics, 160 (2004), 1141–1182 Isomonodromy transformations of linear systems of difference equations By Alexei Borodin Abstract We introduce and study “isomonodromy” transformations of ... general theory of “isomonodromy” transformations for linear systems of difference equations with rational coefficients This subject is of interest in its own right As an application of 1142 ALEXEI ... are matrix elements in A1 , , An 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...

Ngày tải lên: 14/02/2014, 17:20

... Region of a System in Three Unknowns 44 Systems of Linear Inequalities in Any Number of Unknowns 52 The Solution of a System of Linear Inequalities 'by Successive Reduction of the Number of Unknowns ... of this small book to acquaint the reader with the various aspects of the theory of systems of linear inequalities, viz with the geometrical aspect of the matter and some of the methods for solving ... inequalities is in the long run reduced to the solution of a number of systems of linear equations We shall regard the solution of a system of linear equations as something simple, as an elementary operation,...

Ngày tải lên: 08/04/2014, 10:42

... of lyness-type difference equations, ” Advances in Difference Equations, vol 2007, Article ID 31272, 13 pages, 2007 12 B Iri´ anin and S Stevi´ , “Some systems of nonlinear difference equations of ... Papaschinopoulos and C J Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations, ” Communications on Applied Nonlinear Analysis, vol 5, no 2, pp 47–59, 1998 10 ... 10 G Papaschinopoulos and C J Schinas, “Invariants for systems of two nonlinear difference equations, ” Differential Equations and Dynamical Systems, vol 7, no 2, pp 181–196, 1999 11 G Papaschinopoulos,...

Ngày tải lên: 22/06/2014, 11:20

... exercise Resources Solution of sets of linear equations is required in many petroleum applications Many methods exist for this purpose, direct methods as well as iterative methods The reference mentioned ... this method will be discussed below First, let’s review the concept of simultaneous linear equations A set of linear simultaneous equations may be written as: a11 x1 + a12 x2 + a13 x3 + + a1N xN ... bM       x M  If the number of unknowns is equal to the number of equations, N=M, we may be able to solve the set of equations, provided that the equations are unique Gaussian Elimination...

Ngày tải lên: 24/10/2014, 21:28

... define matrix D ∈ R r×r Here Ir and On−r stand for the identity and zero matrices of appropriate dimensions respectively Proof The symmetric and nonnegative matrix A can be diagonalized by an ... of Science, Mathematics - Physics 23 (2007) 201-209 205 Algorithm PFS Step Initialize v := v0h Step For given m ≥ and vm (an approximation of vh (mτ )) find v m+1,k by solving (in parallel) systems ... Discretizing the BVP (12)-(13) one obtains a large-scale system of linear equations Lw = g, (19) where L is a symmetric positive define matrix of dimension p × p, where p = p(h) depends on the discretization...

Ngày tải lên: 13/02/2014, 03:20

Ngày tải lên: 14/03/2014, 13:20

... row sum matrix norm 1, page 295 Define a matrix norm · by M SMS−1 ∞ Then we have A ≤ Let λ be an eigenvalue of a matrix A ∈ Mn The index of λ, denoted by index λ is the smallest value of k for ... SJS−1 , then the k-average of A is SJ S−1 , where J is the k-average of J J is in turn composed of the k-average of each of its Jordan blocks For a Jordan block of J of the form ⎞ λ ⎟ ⎜ ⎟ ⎜ λ ... eigenvalue of B Proof As in the proof of Theorem 2, a and b are equivalent For < μ < 1, denote μI − μ B by B μ Suppose now that a holds Let λ be an eigenvalue of B Then μ 1− μ λ is an eigenvalue of...

Ngày tải lên: 21/06/2014, 20:20

... Biopolymer removal by sole Fe(III), mixtures of polysilicate addition (Error bars show the standard deviation of samples) Biopolymer removal by PSI addition Table shows the results of protein and ... behavior of polysilicate in controlling membrane fouling in three MBR systems (R1, R2, and R3) TMP was increased by membrane clogging and recovered by physical washing The concentrations of sole ... 25 30 35 Time (days) Fig - TMP profiles of MBR at a) R4, b) R5 and c) R6 TMP profiles of MBR experiments with PSI addition Fig shows the fouling frequency in MBR systems when PSI and sole Fe(III)...

Ngày tải lên: 05/09/2013, 10:15

... Much of the sophistication of complicated linear equation -solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... N by the reference a[i] Tasks of Computational Linear Algebra • Solution of the matrix equation A·x = b for an unknown vector x, where A is a square matrix of coefficients, raised dot denotes matrix ... corresponding x’s are then the columns of the matrix inverse of A (§2.1 and §2.3) • Calculation of the determinant of a square matrix A (§2.3) If M < N , or if M = N but the equations are degenerate, then...

Ngày tải lên: 15/12/2013, 04:15

... any row in A by a linear combination of itself and any other row, as long as we the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • ... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... 4) of the vector solution of the jth right-hand side (j = 1, 2, 3), the one whose coefficients are bij , i = 1, 2, 3, 4; and that the matrix of unknown coefficients yij is the inverse matrix of...

Ngày tải lên: 15/12/2013, 04:15

... involve solving a succession of linear systems each of which differs only slightly from its predecessor Instead of doing O(N ) 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 Sample page from NUMERICAL RECIPES IN C: THE ART OF ... We write a Householder matrix in the form − u ⊗ u/c where c = u · u An appropriate Householder matrix applied to a given matrix can zero all elements in a column of the matrix situated below a...

Ngày tải lên: 15/12/2013, 04:15

... the linear set A · x = (L · U) · x = L · (U · x) = b (2.3.3) by first solving for the vector y such that L·y=b (2.3.4) U·x=y (2.3.5) and then solving What is the advantage of breaking up one linear ... this advantage 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, ... Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) where L is lower triangular...

Ngày tải lên: 24/12/2013, 12:16

... and 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, ... America) Determinant of a Matrix 50 Chapter Solution of Linear Algebraic Equations A quick-and-dirty way to solve complex systems is to take the real and imaginary parts of (2.3.16), giving A·x−C·y=b ... 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

... 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 ... obtain greater precision for the solution of a linear set than the precision of your computer’s floating-point word Unfortunately, for large sets of linear equations, it is not always easy to obtain ... solution of the linear system by LU decomposition can be accomplished much faster, and in much less storage, than for the general N × N case The precise definition of a band diagonal matrix with...

Ngày tải lên: 24/12/2013, 12:16

... c’s are scalars, but as matrix inversion if the a’s and c’s are themselves submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively by partitions in half At ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software Permission ... “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)?...

Ngày tải lên: 24/12/2013, 12:16

... 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 as the largest amplification of length that it ... discussion of the use of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond ... x[1 n] of the linear set of equations A · X = B The matrix 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], the LU decomposition of a...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... look at solving the set of simultaneous linear equations (2.6.6) in the case that A is singular First, the set of homogeneous equations, where b = 0, is solved immediately by SVD: Any column of V ... 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 so corrupted by...

Ngày tải lên: 24/12/2013, 12:16