/home/abui.../sgesvdq.f(3)

🌸 NAME 🌸

okay so this is the sgesvdq.f(3) page from the Library Functions Manual, and the source path is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sgesvdq.f

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine SGESVDQ (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u,
    ldu, v, ldv, numrank, iwork, liwork, work, lwork, rwork, lrwork,
    info)

SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

🛠️ Function/Subroutine Documentation 🛠️

subroutine SGESVDQ (character joba, character jobp, character jobr,
    character jobu, character jobv, integer m, integer n, real, dimension(
    lda, * ) a, integer lda, real, dimension( * ) s, real, dimension( ldu, *
    ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, integer
    numrank, integer, dimension( * ) iwork, integer liwork, real, dimension(
    * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork,
    integer info)

and the one-liner is literally: SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices 💅

💖 Purpose 💖

okay bestie, here's the vibe: SGESVDQ computes the singular value decomposition (SVD) of a real M-by-N matrix A, where M >= N. The SVD of A is written as:

                                   [++]   [xx]   [x0]   [xx]
             A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
                                   [++]   [xx]

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements 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. iconic matrix behavior, no cap ✨

🧚 Parameters 🧚

`JOBA`

JOBA is CHARACTER*1. It specifies the level of accuracy in the computed SVD.

'A' — the requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F, where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to truncate the computed triangular factor in a rank revealing QR factorization whenever the truncated part is below the threshold of the order of EPS * ||A||_F. This is the aggressive truncation level 💅
'M' — similarly as with 'A', but the truncation is more gentle: it is allowed only when there is a drop on the diagonal of the triangular factor in the QR factorization. This is the medium truncation level.
'H' — high accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted.
'E' — same as 'H', and in addition the condition number of column scaled A is estimated and returned in RWORK(1). N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1).

`JOBP`

JOBP is CHARACTER*1.

'P' — the rows of A are ordered in decreasing order with respect to ||A(i,:)||_\infty. This enhances numerical accuracy at the cost of extra data movement. Recommended for numerical robustness.
'N' — no row pivoting.

`JOBR`

JOBR is CHARACTER*1.

'T' — after the initial pivoted QR factorization, SGESVD is applied to the transposed R**T of the computed triangular factor R. This involves some extra data movement (matrix transpositions). Useful for experiments, research and development.
'N' — the triangular factor R is given as input to SGESVD. This may be preferred as it involves less data movement.

`JOBU`

JOBU is CHARACTER*1.

'A' — all M left singular vectors are computed and returned in the matrix U. See the description of U.
'S' or 'U' — N = min(M,N) left singular vectors are computed and returned in the matrix U. See the description of U.
'R' — numerical rank NUMRANK is determined and only NUMRANK left singular vectors are computed and returned in the matrix U.
'F' — the N left singular vectors are returned in factored form as the product of the Q factor from the initial QR factorization and the N left singular vectors of (R**T , 0)**T. If row pivoting is used, then the necessary information on the row pivoting is stored in IWORK(N+1:N+M-1).
'N' — the left singular vectors are not computed.

`JOBV`

JOBV is CHARACTER*1.

'A', 'V' — all N right singular vectors are computed and returned in the matrix V.
'R' — numerical rank NUMRANK is determined and only NUMRANK right singular vectors are computed and returned in the matrix V. This option is allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
'N' — the right singular vectors are not computed.

bestie, clock it: JOBV = 'R' is only legal when JOBU = 'R' or JOBU = 'N'. otherwise it is illegal — periodt 💀

`M`

M is INTEGER. The number of rows of the input matrix A. M >= 0.

`N`

N is INTEGER. The number of columns of the input matrix A. M >= N >= 0.

`A`

A is REAL array of dimensions LDA x N.

On entry, the input matrix A.

On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains the Householder vectors as stored by SGEQP3. If JOBU = 'F', these Householder vectors together with WORK(1:N) can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

`S`

S is REAL array of dimension N. The singular values of A, ordered so that S(i) >= S(i+1).

`U`

U is REAL array, dimension:

LDU x M if JOBU = 'A'; see the description of LDU. In this case, on exit, U contains the M left singular vectors.
LDU x N if JOBU = 'S', 'U', 'R'; see the description of LDU. In this case, U contains the leading N or the leading NUMRANK left singular vectors.
LDU x N if JOBU = 'F'; see the description of LDU. In this case U contains N x N orthogonal matrix that can be used to form the left singular vectors.
If JOBU = 'N', U is not referenced.

`LDU`

LDU is INTEGER. The leading dimension of the array U.

If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
If JOBU = 'F', LDU >= max(1,N).
Otherwise, LDU >= 1.

`V`

V is REAL array, dimension:

LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E'.
If JOBV = 'A', or 'V', V contains the N-by-N orthogonal matrix V**T.
If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right singular vectors, stored rowwise, of the NUMRANK largest singular values).
If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
If JOBV = 'N', and JOBA.NE.'E', V is not referenced.

`LDV`

LDV is INTEGER. The leading dimension of the array V.

If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
Otherwise, LDV >= 1.

`NUMRANK`

NUMRANK is INTEGER.

NUMRANK is the numerical rank first determined after the rank revealing QR factorization, following the strategy specified by the value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK leading singular values and vectors are then requested in the call of SGESVD. The final value of NUMRANK might be further reduced if some singular values are computed as zeros.

`IWORK`

IWORK is INTEGER array, dimension (max(1, LIWORK)).

On exit, IWORK(1:N) contains column pivoting permutation of the rank revealing QR factorization.

If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence of row swaps used in row pivoting. These can be used to restore the left singular vectors in the case JOBU = 'F'.

If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, IWORK(1) returns the minimal LIWORK.

`LIWORK`

LIWORK is INTEGER. The dimension of the array IWORK.

LIWORK >= N + M - 1,     if JOBP = 'P' and JOBA .NE. 'E';
LIWORK >= N              if JOBP = 'N' and JOBA .NE. 'E';
LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';
LIWORK >= N + N          if JOBP = 'N' and JOBA = 'E'.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA.

`WORK`

WORK is REAL array, dimension (max(2, LWORK)), used as a workspace.

On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters needed to recover the Q factor from the QR factorization computed by SGEQP3.

If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, WORK(1) returns the optimal LWORK, and WORK(2) returns the minimal LWORK.

`LWORK`

LWORK 💅

bestie, LWORK is an INTEGER — it's literally just the dimension of the WORK array, and the way it's decided is a whole journey. Here's the tea, exactly:

The dimension of the array WORK. It is determined as follows:
Let  LWQP3 = 3*N+1,  LWCON = 3*N, and let
LWORQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
        { MAX( M, 1 ),  if JOBU = 'A'
LWSVD = MAX( 5*N, 1 )
LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),
LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )
Then the minimal value of LWORK is:
= MAX( N + LWQP3, LWSVD )        if only the singular values are needed;
= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
                         and a scaled condition estimate requested;

= N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left
                         singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left
                         singular vectors are requested, and also
                         a scaled condition estimate requested;

= N + MAX( LWQP3, LWSVD )        if the singular values and the right
                         singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
                         singular vectors are requested, and also
                         a scaled condition etimate requested;

= N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R';
                         independent of JOBR;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,
                         JOBV = 'R' and, also a scaled condition
                         estimate requested; independent of JOBR;
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the
               full SVD is requested with JOBV = 'A' or 'V', and
               JOBR ='N'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )
               if the full SVD is requested with JOBV = 'A' or 'V', and
               JOBR ='N', and also a scaled condition number estimate
               requested.
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the
               full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )
               if the full SVD is requested with JOBV = 'A' or 'V', and
               JOBR ='T', and also a scaled condition number estimate
               requested.
Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).

If LWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.

So basically: pick your JOB* options, and one of these formulas picks your minimum LWORK for you. And yeah — that last line! If LWORK = -1 you're in workspace-query mode: the routine just computes and hands back the optimal + minimal sizes for the WORK, IWORK, and RWORK arrays, and XERBLA will not yell at you about LWORK. ✨

RWORK 🎀

RWORK is a REAL array, dimension (max(1, LRWORK)). On exit it's giving output-notes:

If JOBA = 'E', then RWORK(1) contains an estimate of the condition number of the column-scaled A. If A = C * D where D is diagonal and C has unit columns in the Euclidean norm, then — assuming full column rank — N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1). Otherwise, RWORK(1) = -1. 🫶
RWORK(2) contains the number of singular values computed as exact zeros in SGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In the case of an early exit (no call to SGESVD, like when the matrix is zero) RWORK(2) = -1.

And the plot twist: if LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, RWORK(1) returns the minimal LRWORK. 🌸

LRWORK 💖

LRWORK is an INTEGER. It's the dimension of the array RWORK.

If JOBP = 'P', then LRWORK >= MAX(2, M).
Otherwise, LRWORK >= 2.

And same energy as before: if LRWORK = -1, then a workspace query is assumed — the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA.

INFO ⭐

INFO is an INTEGER, and it's the vibe check on your results:

= 0: successful exit. slay. 👑
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if SBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B (computed in SGESVD) did not converge to zero. 💀

✨ Further Details ✨

The data movement (matrix transpose) is coded using simple nested DO-loops because BLAS and LAPACK do not provide corresponding subroutines. Those DO-loops are easily identified in this source code — by the CONTINUE statements labeled with 11**. In an optimized version of this code, the nested DO loops should be replaced with calls to an optimized subroutine.
This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause column norm overflow. This is the minial precaution and it is left to the SVD routine (CGESVD) to do its own preemptive scaling if potential over- or underflows are detected. To avoid repeated scanning of the array A, an optimal implementation would do all necessary scaling before calling CGESVD and the scaling in CGESVD can be switched off.
Other comments related to code optimization are given in comments in the code, enclosed in [[double brackets]]. 🦋

🐛 Bugs, examples and comments 🙈

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you. (ngl they're so polite about it 🥹)

📚 References 📚

[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for Computing the SVD with High Accuracy. ACM Trans. Math. Softw. 44(1): 11:1-11:30 (2017)

SIGMA library, xGESVDQ section updated February 2016. Developed and coded by Zlatko Drmac, Department of Mathematics University of Zagreb, Croatia, drmac@math.hr

🌈 Contributors 🌈

Developed and coded by Zlatko Drmac, Department of Mathematics University of Zagreb, Croatia, drmac@math.hr

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 412 of file sgesvdq.f.

and that's the whole page, bestie — generated automatically by Doxygen for LAPACK from the source code, LAPACK Version 3.12.0, dated /home/abui.../sgesvdq.f(3). go off, you understood all of it, i'm obsessed 🫶💅✨

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