Introduction to Sparse Matrices In Scilab - Projects

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lufactlusolvelugetludelchfactchsolvegmrespcgqmrsparse lu factorizationsparse linear system solverextraction of sparse LU factorsutility function used with lufactsparse Cholesky factorizationsparse Cholesky solverGeneralized Minimum RESidual methodprecondioned conjugate gradientquasi minimal resiqual method with preconditioningFigure 2 – Direct and iterative functions to solve sparse linear equations.3 Solving sparse linear equationsScilab provide direct and iterative methods to solve linear systems ofequations. The figure 2 presents these methods.3.1 Sparse LU decompositionThe sparse LU decomposition available in Scilab is based on the Sparsepackage written by Kenneth S. Kundert and Alberto Sangiovanni-Vincentelli[4]. This package is available in Netlib [3].Sparse is a flexible package of subroutines written in C used to quicklyand accurately solve large sparse systems of linear equations. The package isable to handle arbitrary real and complex square matrix equations. Besidesbeing able to solve linear systems, it is also able to quickly solve transposedsystems, find determinants, and estimate errors due to ill-conditioning in thesystem of equations and instability in the computations. Sparse also providesa test program that is able read matrix equations from a file, solve them, andprint useful information about the equation and its solution.Sparse is generally as fast or faster than other popular sparse matrixpackages when solving many matrices of similar structure. Sparse does notrequire or assume sym- metry and is able to perform numerical pivoting toavoid unnecessary error in the solution. It handles its own memory allocation,which allows the user to forgo the hassle of providing adequate memory. Italso has a natural, flexible, and efficient interface to the calling program.Sparse was originally written for use in circuit simu- lators and is parti-10
cularly apt at handling node- and modified-node admittance matrices. Thesystems of linear generated in a circuit simulator stem from solving largesystems of nonlinear equations using Newton’s method and integrating largestiff systems of ordinary differential equations. However, Sparse is also suitablefor other uses, one in particular is solving the very large systems oflinear equations resulting from the numerical solution of partial differentialequations.The lufact, lusolve, luget and ludel functions is a set of functionsproviding a sparse direct method to solve linear systems of equations. Astheir names suggest it, they use a sparse LU decomposition to compute thesolution.In the following script, we define a sparse matrix spA with entries (1, 2, 3, 4)on the diagonal only. Then we use the lufact function to compute the LUdecomposition of the matrix spA. The output of the lufact function are ahandle to the factors h and the rank rk of the matrix. The variable h has nodirect use for the user. In fact, it represents a memory location where Scilabhas stored the actual factors. The purpose of the variable h is to be an inputargument of the lusolve, luget and ludel functions. In the script, we passh and the right hand-side b to the lusolve function, which produces thesolution x of the equation Ax = b. Finally, we delete the LU factors with theludel function.non_zeros =[1 ,2 ,3 ,4];rows_cols =[1 ,1;2 ,2;3 ,3;4 ,4];spA = sparse ( rows_cols , non_zeros )[h,rk] = lufact (sp );b = [1 1 1 1] ’;x= lusolve (h,b)ludel (h);The previous script produces the following output.--> non_zeros =[1 ,2 ,3 ,4];--> rows_cols =[1 ,1;2 ,2;3 ,3;4 ,4];-->spA = sparse ( rows_cols , non_zeros )spA =( 4, 4) sparse matrix( 1, 1) 1.( 2, 2) 2.( 3, 3) 3.( 4, 4) 4.-->[h,rk] = lufact (sp );-->b = [1 1 1 1] ’;11
Page 2: 4 Cholesky factorizations 155 Funct
Page 6 and 7: sp2adjspeyesponessprandspzerosfulls
Page 8 and 9: -->D= sparse (C)D =( 3, 5) sparse m
Page 12 and 13: -->x= lusolve (h,b)x =1.0.50.333333
Page 14 and 15: 0 0 0 0 28 20 71 39 0 00 10 0 0 32
Page 16 and 17: PlotSparseReadHBSparsecond2spcondes
Page 18 and 19: umf licenseumf ludelumf lufactumf l
Page 20 and 21: Read sparse doubles matrices in gat
Page 22 and 23: int read_sparse ( char * fname , un
Page 24 and 25: dnaupddneupddsaupddsaupdznaupdzneup
Page 26 and 27: Information about MatrixMarket file
Page 28 and 29: Sparse matrices can be managed in S
Page 30 and 31: The solver argument is designed so
Page 32 and 33: function u = mysolverTAUCS (N,b)A =
Page 34 and 35: -->A = scibench_poissonA ( 500 );--

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