Leslie Foster. Department of Mathematics ... LESLIE FOSTER DEPARTMENT OF MATHEMATICS, SAN JOSE STATE UNIVERSITY ... SPQR (Tim Davis, 2009):.
CALCULATING RANKS, NULL SPACES AND PSEUDOINVERSE SOLUTIONS FOR SPARSE MATRICES USING SPQR Leslie Foster Department of Mathematics, San Jose State University
October 28, 2009, SIAM LA 09
L ESLIE F OSTER D EPARTMENT OF M ATHEMATICS , S AN J OSE S TATE U NIVERSITY CALCULATING RANKS, NULL SPACES AND PSEUDOINVER
O UTLINE Goal: Reliable algorithm for calculating ranks, null spaces and pseudoinverse solutions for large sparse matrices in the presence of errors I. II. III. IV. V.
Numerical rank, Numerical Null Space, . . . Tools: SPQR, subspace iteration Algorithm Numerical Experiments Conclusions
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
N UMERICAL R ANK , N UMERICAL N ULL S PACE
For m × n matrix A and a tolerance tol Numerical rank: r = no. of singular values > tol Numerical Null Space Basis: n × (n − r ) matrix with orthonormal columns with ||AX|| tol T References: Chan, Vogel, Gotsman, Toledo L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
A LGORITM SPQR_NULL
SPQR_NULL: returns accurate numerical rank, orthogonal null space basis, and pseudoinverse solution “accurate numerical rank” means that when the estimated numerical rank is not correct, a flag is returned warning the user
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
A LGORITHM SPQR_NULL, CONT ’ D Input: m × n matrix A and tolerance tol Use SPQR twice to construct approximate complete orthogonal decomposition b 0 T A = Q1 P2 Q2T P1T + E, 0 0 with Q1 , Q2 , P1 , P2 orthogonal. Know ||E||F . b for numerical null Use SSI to find ⊥ basis X b. T b usually has a small, often 0, space of T nullity. L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
A LGORITHM SPQR_NULL, CONT ’ D The ⊥ nullspace basis of A is b 0 X . Save in factored form. X = P1 Q2 0 I Use singular value estimates and ||E||F to confirm that the rank is correct, or return warning if not possible to confirm. b is Use ⊥ factorization and, when T numerical singular, deflation to calculate pseudoinverse solution. L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
M ATRIX T EST S ET SJSU Singular Matrix Database at http://www.math.sjsu.edu/ singular/matrices/ 767 numerically singular matrices (with tolerance max(m, n) eps(normest(A)) ) Matrices come from real world applications or have characteristic features of real world problems Most matrices from UF Sparse Matrix Collection (Davis) L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
P ROPERTIES OF T EST S ET M ATRICES :
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
P ROPERTIES OF T EST S ET M ATRICES ( CONT ’ D ):
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
ACCURACY, CORRECT RANKS :
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
ACCURACY, CORRECT RANKS OR WARNING :
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
ACCURACY OF R ANK C ALCULATION :
warning flag had no false positives L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
ACCURACY OF N ULL S PACE BASES :
QR based method overall as good as SVD L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
RUN T IMES : SPQR_NULL VERSUS MATLAB’ S SVD:
L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
RUN T IMES : SPQR_NULL VERSUS MATLAB’ S SVDS: For a run over 372 matrices with smaller dimensions MATLAB’s SVDS required 8 hours SPQR_NULL required 1.5 minutes SVDS error flag reported an error for 89% of the matrices MATLAB’s SVDS not useful for finding null spaces bases of matrices L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
RUN T IMES : SPQR_NULL VERSUS SPQR:
SPQR_NULL is slower than SPQR L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
M EMORY U SE FOR N ULL S PACE BASES
Overall SPQR_NULL null space bases requires less memory than dense bases L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
C ONCLUSIONS / F URTHER WORK : SPQR_NULL reliably determines rank, null space basis and psuedoinverse solutions Much faster than MATLAB’s SVD More reliable than SPQR and MATLAB’s SVDS More features than SPQR Further work: Related algorithms Applications L ESLIE F OSTER
SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
RUN T IMES : SPQR_NULL VERSUS SPQR + SSI
maximum number total time total time numerical of (hours) for (hours) for nullity matrices SPQR_NULL SPQR + SSI 500 479 6.7 4.2 1000 517 8.5 16.6 1500 550 10.8 42.7 394336 738 21.8 not available TABLE : Total times (sum of the run times for the matrices in the indicated subsets of our test set) for SPQR_NULL and SPQR + SSI
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SIAM L INEAR A LGEBRA C ONFERENCE 2009, O CT. 26-31, 2009, M
