solution of a system of linear equations by matrix inversion method

... elimination method Gaussian elimination procedure Programming exercise Make a FORTRAN program consisting of a main program, that reads the coefficients of the system of equations (n, a1 ,1…an,n ... Objectives The equations may alternatively be written in a compact form: Introduction Gaussian elimination method Gaussian elimination procedure A x = b where A is the coefficient matrix, and b is ... 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 Method Info FAQ Reference...

Ngày tải lên: 24/10/2014, 21:28

... 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 multiplication, and b is a known ... many cases you will have no alternative but to use sophisticated black-box program packages Several good ones are available, though not always in C LINPACK was developed at Argonne National Laboratories ... call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations...

Ngày tải lên: 15/12/2013, 04:15

... or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) a1 1  a2 1 a3 1 a4 1 38 Chapter Solution of Linear Algebraic Equations Pivoting Sample page ... 37 2.1 Gauss-Jordan Elimination Elimination on Column-Augmented Matrices  Consider the linear matrix equation      a1 2 a2 2 a3 2 a4 2 a1 3 a2 3 a3 3 a4 3 a1 4 x11 a2 4   x21  · a3 4 x31 a4 4 x41 ... Data Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill), Program B-2, p 298 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations...

Ngày tải lên: 15/12/2013, 04:15

... 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 chosen ... scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to solve the next set of linear equations The LU decomposition is complicated to update ... new upper triangular matrix R The matrix Q is 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...

Ngày tải lên: 15/12/2013, 04:15

... The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution over Gauss-Jordan elimination is ... Gaussian elimination thus has about a factor three advantage over Gauss-Jordan (We could reduce this advantage to a factor 1.5 by not computing the inverse matrix as part of the Gauss-Jordan scheme.) ... Numerical Analysis, 2nd ed (Reading, MA: AddisonWesley), §2.2.1 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able...

Ngày tải lên: 24/12/2013, 12:16

... 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, that is, ... 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 (2.3.17) C·x +A y=d ... 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 and the sign separately Complex Systems of Equations...

Ngày tải lên: 24/12/2013, 12:16

... Solution of Linear Algebraic Equations In that case, the 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 ... referenced by any manipulations and can have arbitrary values Notice that the diagonal of the original matrix appears in column m1 + 1, with subdiagonal elements to its left, superdiagonal elements ... that its nonzero elements lie in a long, narrow matrix with m1 + + m2 columns and N rows This is best illustrated by an example: The band diagonal matrix 2.4 Tridiagonal and Band Diagonal Systems...

Ngày tải lên: 24/12/2013, 12:16

... this “fast” matrix multiplication, Strassen also obtained a surprising result for matrix inversion [1] Suppose that the matrices a1 1 a2 1 a1 2 a2 2 and c11 c21 c12 c22 (2.11.5) are inverses of each ... 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 each step, halving the order ... to trade@cup.cam.ac.uk (outside North America) In (2.11.6) the “inverse” operator occurs just twice It is to be interpreted as the reciprocal if the a s and c’s are scalars, but as matrix inversion...

Ngày tải lên: 24/12/2013, 12:16

... 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 as the largest amplification of length ... fact that the LU decomposition of A is itself not exact A different analytical approach starts with some matrix B0 that is assumed to be an approximate inverse of the matrix A, so that B0 · A ... mprove(float * *a, float **alud, int n, int indx[], float b[], float x[]) Improves a solution vector 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...

Ngày tải lên: 24/12/2013, 12:16

... nullspace and range are important Consider the familiar set of simultaneous equations A x=b (2.6.6) where A is a square matrix, b and x are vectors Equation (2.6.6) defines A as a linear mapping ... svbksb for evaluating equation (2.6.7) and obtaining a solution vector x from a right-hand side b, given that the SVD of a matrix A has already been calculated by a call to svdcmp Note that this routine ... singular value corresponds to 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...

Ngày tải lên: 24/12/2013, 12:16

... substantial collection of routines for sparse matrix calculation is also available from IMSL [5] as the Yale Sparse Matrix Package [6] You should be aware that the special order of interchanges and ... in the matrix, or the number of elements in the arrays sa and ija.) Location N + of sa is not used and can be set arbitrarily • Entries in sa at locations ≥ N + contain A s off-diagonal values, ... trade@cup.cam.ac.uk (outside North America) zeros 73 2.7 Sparse Linear Systems Sherman-Morrison Formula Suppose that you have already obtained, by herculean effort, the inverse matrix A 1 of a...

Ngày tải lên: 24/12/2013, 12:16

... about a factor of two faster than alternative methods for solving linear equations Instead of seeking arbitrary lower and upper triangular factors L and U, Cholesky decomposition constructs a ... diagonal principal minors of the original Toeplitz matrix vanish (Compare with discussion of the tridiagonal algorithm in §2.4.) If the algorithm fails, your matrix is not necessarily singular ... Chapter Solution of Linear Algebraic Equations Papers by Bunch [6] and de Hoog [7] will give entry to the literature CITED REFERENCES AND FURTHER READING: Golub, G.H., and Van Loan, C.F 1989, Matrix...

Ngày tải lên: 21/01/2014, 18:20

... 90 Chapter Solution of Linear Algebraic Equations 2.8 Vandermonde Matrices and Toeplitz Matrices Vandermonde Matrices A Vandermonde matrix of size N × N is completely determined by N arbitrary ... or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) In §2.4 the case of a tridiagonal matrix was treated specially, because that particular ... about a factor of two faster than alternative methods for solving linear equations Instead of seeking arbitrary lower and upper triangular factors L and U, Cholesky decomposition constructs a...

Ngày tải lên: 21/01/2014, 18:20

... Numerical Analysis, Brooks/Cole, Pacific Grove, Calif, USA, 1991 S Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings of the American Mathematical Society, ... matrix with aii / for all i 1, , n Let Q D or L, where D is the diagonal matrix containing the diagonal of A, and L the lower triangular matrix containing the lower triangular entries of A ... dominant with aii / for every i and if Q D or L, where D is the diagonal matrix containing the diagonal of A, and L the lower triangular matrix containing the lower triangular entries of A, then...

Ngày tải lên: 21/06/2014, 20:20

... STIFFNESS MATRIX OF THE BEAM ON THE ELASTIC FOUNDATION IN THE SYSTEM OF THE GLOBAL CO-ORDINATE In the above part, we presented the stiffness matrix with the system of local co-ordinate of element ... making the calculation we have to transform this matrix to the global coordinate Figure presents the cant bar element with any angle β of horizontal axis x Displacement is presented by two system ... the parapol grade with span 2b and height hv: y= x2 b.f kc In which: Figure 4.The collapse diagram of soil b : a haft of span arch around tunnel structure f kc : strong coefficient At that time,...

Ngày tải lên: 22/07/2014, 03:20

... S, Yamanoi A, Minari Y, Kawabata Y, Ueda S, Hira E, Yamamoto T, Nishi T, Hyakudoi R, Inao T: Reconstruction of the Gastrointestinal Tract by Hemi-Double Stapling Method for the Esophagus and ... via a trocar at the left abdomen (Figure 4, a) A tube was led outside the abdominal cavity while the valve tip at a small hole at the open end of the esophagus was being grasped The cuffs of ... dissection of the gastric lymph nodes, an automated stapler was inserted via a trocar of the right abdomen, and tissues of the esophagus were cut off (Figure 2) A 7-cm-long small abdominal incision was...

Ngày tải lên: 09/08/2014, 01:24

... Nurkanovi´ , Asymptotic behavior of a two dimensional linear fractional c c system of difference equations, Radovi Matematiˇ ki 11 (2002), no 1, 59–78 c , Asymptotic behavior of a system of linear ... E is an attractor and ᏷ is a subset of the basin of attraction of E If F is differentiable at a fixed point E, and if the Jacobian JF (E) has one eigenvalue with modulus less than one and a second ... B,1 − A) of system (1.2) separates the positive quadrant into basins of attraction of two types of asymptotic behavior From a biological perspective, W s ((1 − B,1 − A) ) is a threshold manifold...

Ngày tải lên: 22/06/2014, 22:20

... became aware of a recent paper [10] by Daud´ and e Ravelomanana They studied a close but different case, when a system of m equations is chosen uniformly at random among all n(n − 1) equations of ... Gamarnik, M T Hajiaghayi and G B Sorkin, Random MAX SAT, random MAX CUT, and their phase transitions, Random Structures and Algorithms, 24 (2004) 502–545 [9] N Creignon and H Daud´, Smooth and ... bounded away from both and iff m + O(n2/3 ) A natural extension of the system (1.0.1) is a system of k -linear equations xi = be (mod 2), (1.0.13) i∈e where e runs over a set E of (hyper)edges of a k-uniform...

Ngày tải lên: 08/08/2014, 12:22

... et al Critical Care 1998, 2:79 http://ccforum.com number of bacteria after hand-washing/the number of bacteria before hand-washing)] Values are calculated from raw data and expressed as mean ... Each value was analysed using Mann-Whitney (between two groups) or anlaysis of variance (ANOVA; among three groups) tests Statistical significance was considered at P < 0.05 The number of bacteria ... that of the conventional, alcohol lotion showed significantly higher rates of bacterial removal than that of hand-washing with tap water (52.1 ± 11.3%; P < 0.05) Nosocomial infections are a major...

Ngày tải lên: 12/08/2014, 18:20

... higher than for the MERS case The influence of actual switch characteristics and thermal capability has been investigated An average peak junction temperature of 125°C and 80°C heat sink temperature ... Ryuichi Shimada, Jan A Wiik, Takanori Isobe, Taku Takaku, Noriyuki Iwamuro, Yoshiyuki Uchida, Marta Molinas, and Tore M Undeland A new ac current switch called mers with low on-state voltage igbts ... Relative supply rating 8.4 pu 10.4 pu Rating and loss considerations for large scale system The losses and ratings of a large scale electrical conversion system with MERS solution and with a 2-level...

Ngày tải lên: 15/10/2013, 16:11