solving sets of linear algebraic equations with mathematica

... 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, ... of Linear Algebraic Equations Coleman, T.F., and Van Loan, C 1988, Handbook for Matrix Computations (Philadelphia: S.I.A.M.) Forsythe, G.E., and Moler, C.B 1967, Computer Solution of Linear Algebraic ... 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 for one or more right-hand side vectors...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... 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) • Interchanging ... North America) a11  a21 a31 a41 38 Chapter Solution of Linear Algebraic Equations Pivoting Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...

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 ... to solve the next set of linear equations The LU decomposition is complicated to update because of pivoting However, QR turns out to be quite simple for a very common kind of update, A → A+s⊗t ... simply the product of Q with the 2(N − 1) Jacobi rotations In applications we usually want QT , and the algorithm can easily be rearranged to work with this matrix instead of with Q Sample page...

Ngày tải lên: 15/12/2013, 04:15

... Westlake, J.R 1968, A 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) ... 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 ... 42 Chapter Solution of Linear Algebraic Equations Backsubstitution But how we solve for the x’s? The last x (x4 in this example)...

Ngày tải lên: 24/12/2013, 12:16

... 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; int n,*indx; ... subsequently want to solve a set of equations with the same A but a different right-hand side b, you repeat only lubksb(a,n,indx,b); not, of course, with the original matrix A, but with a and indx as were ... 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

... 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 ... Unfortunately, for large sets of linear equations, it is not always easy to obtain precision equal to, or even comparable to, the computer’s limit In direct methods of solution, roundoff errors accumulate, ... storage space The following routine, bandec, is the band-diagonal analog of ludcmp in §2.3: 54 Chapter Solution of Linear Algebraic Equations #define SWAP(a,b) {dum=(a);(a)=(b);(b)=dum;} void banbks(float...

Ngày tải lên: 24/12/2013, 12:16

... 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, ... “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)? ... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 is able to induce ... 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 ... trade@cup.cam.ac.uk (outside North America) There exists a very powerful set of techniques for dealing with sets of equations or matrices that are either singular or else numerically very close...

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 ... reciprocals 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...

Ngày tải lên: 24/12/2013, 12:16

... applicable to some general classes of sparse matrices, and which not necessarily depend on details of the pattern of sparsity 74 Chapter Solution of Linear Algebraic Equations (A + u ⊗ v) · x = b ... applications.) • Each of the first 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 ... choice 76 Chapter Solution of Linear Algebraic Equations Here A is, as usual, an N × N matrix, while U and V are N × P matrices with P < N and usually P N The inner piece of the correction term...

Ngày tải lên: 24/12/2013, 12:16

... Solution of Linear Algebraic Equations The only remaining problem is to develop a recursion relation for G Before we that, however, we should point out that there are actually two distinct sets of ... 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 ... square root” of the matrix A The components of LT are of course related to those of L by LT = Lji ij (2.9.3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations...

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 ... Solution of Linear Algebraic Equations The only remaining problem is to develop a recursion relation for G Before we that, however, we should point out that there are actually two distinct sets of ... 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...

Ngày tải lên: 21/01/2014, 18:20

Ngày tải lên: 22/03/2014, 11:20

... Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical Sciences University of ... set of real numbers set of nonnegative real numbers d-dimensional Euclidean space Ω sample space ∅ empty set A∪B the union of sets A and B A∩B the intersection of sets A and B A\B the set A without ... approximation of continuous solutions of SDEs The discrete time approximation of SDEs with jumps represents the focus of the monograph The reader learns about powerful numerical methods for the solution of...

Ngày tải lên: 19/02/2014, 22:20

Ngày tải lên: 31/03/2014, 15:56

... G coincides with the Lie algebra of the derivations of L/k that commute with the derivation on L (3) The field LG of G-invariant elements of L is equal to k Proof An intuitive proof of (1) and ... induces a k -linear derivation on R commuting with The latter extends uniquely to a k -linear derivation of L commuting with One can also start with a k -linear derivation of L commuting with and ... of F form a basis of the solution space V (3) Let L denote the field of fractions of R Then one can also consider the group Gal(L/k) consisting of the k -linear automorphisms of L, commuting with...

Ngày tải lên: 31/03/2014, 15:56

... 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: ... A∗ Xδi Ai = Q Linear Algebra Appl 429, 110–121 i=1 i (2008) doi:10.1016/j.laa.2008.02.014 Duan, X, Peng, Z, Duan, F: Positive defined solution of two kinds of nonlinear matrix equations Surv ... 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

... neutral nonlinear differential equations with delay Nonlinear Anal 74(12):3926–3933 (2011) doi:10.1016/j.na.2011.02.029 17 Raffoul, YN: Stability in neutral nonlinear differential equations with functional ... differential equations with variable delays Nonlinear Anal 63, e233–e242 (2005) doi:10.1016/j.na.2005.02.081 11 Sakthivel, R, Luo, J: Asymptotic stability of nonlinear impulsive stochastic differential equations ... differential equations with variable delays Proc Am Math Soc 136, 909–918 (2008) 19 Ardjouni, A, Djoudi, A: Fixed points and stability in linear neutral differential equations with variable delays Nonlinear...

Ngày tải lên: 20/06/2014, 22:20

... Control of systems with aftereffect In: Translations of Mathematical Monographs, vol 157,American Mathematical Society, Providence, RI (1996) Xu, DY: Asymptotic behavior of nonlinear difference equations ... stability of the impulsive difference equations with distributed delays are obtained The conditions (A1)-(A5) are conservative For example, we get the absolute value of all coefficients of (2) We ... impulsive difference equations with distributed delays By establishing an impulsive delay difference inequality and using the properties of “r-cone” and eigenspace of the spectral radius of non-negative...

Ngày tải lên: 21/06/2014, 03:20