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authorjulie <julielangou@users.noreply.github.com>2011-09-30 18:34:50 +0000
committerjulie <julielangou@users.noreply.github.com>2011-09-30 18:34:50 +0000
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* of SIGMA are the singular values of A. The columns of U and V are the
* left and the right singular vectors of A, respectively.
*
-* Further Details
-* ~~~~~~~~~~~~~~~
-* The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
-* rotations. The rotations are implemented as fast scaled rotations of
-* Anda and Park [1]. In the case of underflow of the Jacobi angle, a
-* modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
-* column interchanges of de Rijk [2]. The relative accuracy of the computed
-* singular values and the accuracy of the computed singular vectors (in
-* angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
-* The condition number that determines the accuracy in the full rank case
-* is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
-* spectral condition number. The best performance of this Jacobi SVD
-* procedure is achieved if used in an accelerated version of Drmac and
-* Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
-* Some tunning parameters (marked with [TP]) are available for the
-* implementer.
-* The computational range for the nonzero singular values is the machine
-* number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
-* denormalized singular values can be computed with the corresponding
-* gradual loss of accurate digits.
-*
-* Contributors
-* ~~~~~~~~~~~~
-* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
-*
-* References
-* ~~~~~~~~~~
-* [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
-* SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
-* [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
-* singular value decomposition on a vector computer.
-* SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
-* [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
-* [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
-* value computation in floating point arithmetic.
-* SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
-* [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
-* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
-* LAPACK Working note 169.
-* [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
-* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
-* LAPACK Working note 170.
-* [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
-* QSVD, (H,K)-SVD computations.
-* Department of Mathematics, University of Zagreb, 2008.
-*
-* Bugs, Examples and Comments
-* ~~~~~~~~~~~~~~~~~~~~~~~~~~~
-* Please report all bugs and send interesting test examples and comments to
-* drmac@math.hr. Thank you.
-*
* Arguments
* =========
*
@@ -251,6 +200,60 @@
* of sweeps. The output may still be useful. See the
* description of WORK.
*
+* Further Details
+* ===============
+*
+* The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
+* rotations. The rotations are implemented as fast scaled rotations of
+* Anda and Park [1]. In the case of underflow of the Jacobi angle, a
+* modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
+* column interchanges of de Rijk [2]. The relative accuracy of the computed
+* singular values and the accuracy of the computed singular vectors (in
+* angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
+* The condition number that determines the accuracy in the full rank case
+* is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
+* spectral condition number. The best performance of this Jacobi SVD
+* procedure is achieved if used in an accelerated version of Drmac and
+* Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
+* Some tunning parameters (marked with [TP]) are available for the
+* implementer.
+* The computational range for the nonzero singular values is the machine
+* number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
+* denormalized singular values can be computed with the corresponding
+* gradual loss of accurate digits.
+*
+* Contributors
+* ============
+*
+* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
+*
+* References
+* ==========
+*
+* [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
+* SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
+* [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
+* singular value decomposition on a vector computer.
+* SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
+* [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
+* [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
+* value computation in floating point arithmetic.
+* SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
+* [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
+* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
+* LAPACK Working note 169.
+* [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
+* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
+* LAPACK Working note 170.
+* [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
+* QSVD, (H,K)-SVD computations.
+* Department of Mathematics, University of Zagreb, 2008.
+*
+* Bugs, Examples and Comments
+* ===========================
+* Please report all bugs and send interesting test examples and comments to
+* drmac@math.hr. Thank you.
+*
* =====================================================================
*
* .. Local Parameters ..