RRQR Factorization Linux and Windows MEX-Files for MATLAB
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1 Documentation RRQR Factorization Linux and Windows MEX-Files for MATLAB March 29, Contents of the distribution file The distribution file contains the following files: rrqrgate.dll: the Windows-MEX-File; rrqrgate.mexglx: the Linux-MEX-file; rrqr.m: a Matlab-Script that calls the gateway; matlabblas: a BASH-Script to change the used BLAS-implementation MAT- LAB uses (only Linux); Readme.pdf: this documentation. 2 Usage 2.1 Usage of the gateway The rrqr.m gateway accepts input and supplies output as described in the following. [Q,R, p,r] = rrqr(a), where A is m-by-n, produces an m-by-n upper triangular matrix R and an m-by-m unitary matrix Q so that [ ] R11 R A P = Q 12, 0 R 22 p is a permutation vector and r is the rank of A. 1
2 [Q, R, p, r] = rrqr(a, s ) produces the economy size decomposition. If m n, R is m-by-n and Q is m-by-m, otherwise R is n-by-n and Q is m-by-n. [Q,R, p,r] = rrqr(a,tol). 1 tol specifies an upper bound on the condition number of R 11. If tol == 0 or tol is unset, tol = machine precision is chosen as default. tol must be 0. The tol parameter can be combined with the s parameter. [B,R, p,r] = rrqr(a,c) returns a matrix B so that B = C Q. The tol parameter is accepted as well. [R, p,r] = rrqr(a) is identical to the upper cases but does not compute Q. The tol and s parameters are accepted as well. 2.2 Usage of the BASH-Script matlabblas To change the BLAS-library Mathlab uses, set the appropriate path in the script an run it from a BASH-shell. It will set two variables and call Matlab. So make sure, the matlab-command is in your path. 3 Measurements and comparisons of computing times 3.1 Measurements on Athlon-MP CPUs The results shown in the following sections are obtained from 100 runs with random matrices of different size and rank on a SMP workstation with two Athlon-MP processors using Gentoo Linux and Matlab 7 R14 SP1. (eqs) means the economy size decomposition of the matrices Full rank 1000x1000 random matrix Table shows results which are obtained from 100 runs with random 1000x1000 matrices of full rank. 2
3 Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 1: Results using full rank 1000x1000 matrices on AMD Athlon x1000 random matrix of rank 250 The same test as above is shown in table 2, but now with random 1000x1000 matrices of rank 250. Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 2: Results using 1000x1000 matrices of rank 250 on AMD Athlon. 3
4 x500 random matrix of rank 100 Table 3 shows the results of 100 runs with 1000x500 matrices of rank 100. Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 3: Results using 1000x500 matrices of rank 100 on AMD Athlon x1000 random matrix of rank 100 Table 4 shows the results of 100 runs with 500x1000 matrices of rank
5 Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 4: Results using 500x1000 matrices of rank 100 on AMD Athlon. 3.2 Measurements on an Intel Pentium M CPU The results shown in the following sections are obtained from 100 runs with random matrices of different size and rank on a laptop-computer with an Intel Pentium M processor at 2GHz, using Gentoo Linux and Matlab 7 R14 SP1. (eqs) means the economy size decomposition of the matrices Full rank 1000x1000 random matrix Table 5 shows results which are obtained from 100 runs with random 1000x1000 matrices of full rank. 5
6 Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 5: Results using full rank 1000x1000 matrices on Intel Pentium M x1000 random matrix of rank 250 The same test as above is shown in table 6, but now with random 1000x1000 matrices of rank 250. Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 6: Results using 1000x1000 matrices of rank 250 on Intel Pentium M. 6
7 x500 random matrix of rank 100 Table 7 shows the results of 100 runs with 1000x500 matrices of rank 100. Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 7: Results using 1000x500 matrices of rank 100 on Intel Pentium M x1000 random matrix of rank 100 Table 8 shows the results of 100 runs with 500x1000 matrices of rank
8 Method Mean value [s] Min. [s] Max. [s] Std deviation [s] Variance [s 2 ] svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) svd svd (eqs) qr qr (eqs) rrqr rrqr (eqs) Table 8: Results using 500x1000 matrices of rank 100 on Intel Pentium M. 8
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