gejsv(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

okay bestie, welcome to gejsv(3) from the Library Functions Manual — this is the high-level Jacobi SVD page, and it's a whole saga. The name situation is literally: gejsv — gejsv: SVD, Jacobi, high-level. 💅

✨ SYNOPSIS ✨

🌸 Functions 🌸

subroutine cgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda,
    sva, u, ldu, v, ldv, cwork, lwork, rwork, lrwork, iwork, info)
    CGEJSV
subroutine dgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda,
    sva, u, ldu, v, ldv, work, lwork, iwork, info)
    DGEJSV
subroutine sgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda,
    sva, u, ldu, v, ldv, work, lwork, iwork, info)
    SGEJSV
subroutine zgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda,
    sva, u, ldu, v, ldv, cwork, lwork, rwork, lrwork, iwork, info)
    ZGEJSV

🛠️ Detailed Description 🛠️

okay so this is the high-level detail zone — the page then rolls straight into Function Documentation. No extra side quests here, bestie. 🫶

🧚 Function Documentation 🧚

Here's the full cgejsv signature, exactly as printed — do not freestyle the types, bestie. 💅

subroutine cgejsv (character*1 joba, character*1 jobu, character*1 jobv,
    character*1 jobr, character*1 jobt, character*1 jobp, integer m, integer
    n, complex, dimension( lda, * ) a, integer lda, real, dimension( n )
    sva, complex, dimension( ldu, * ) u, integer ldu, complex, dimension(
    ldv, * ) v, integer ldv, complex, dimension( lwork ) cwork, integer
    lwork, real, dimension( lrwork ) rwork, integer lrwork, integer,
    dimension( * ) iwork, integer info)
    CGEJSV

🎀 Purpose 🎀

CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N matrix [A], where M >= N. The SVD of [A] is written as:

[A] = [U] * [SIGMA] * [V]^*,

where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and [V] is an N-by-N unitary 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. The matrices [U] and [V] are computed and stored in the arrays U and V, respectively. The diagonal of [SIGMA] is computed and stored in the array SVA. 💖

🌸 Parameters 🌸

💅 JOBA 💅

JOBA is CHARACTER*1. It specifies the level of accuracy: ✨

  • 'C' — This option works well (high relative accuracy) if A = B * D, with well-conditioned B and arbitrary diagonal matrix D. The accuracy cannot be spoiled by COLUMN scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is bounded by f(M,N)*epsilon* cond(B), independent of D. The input matrix is preprocessed with the QRF with column pivoting. This initial preprocessing and preconditioning by a rank revealing QR factorization is common for all values of JOBA. Additional actions are specified as follows:
  • 'E' — Computation as with 'C' with an additional estimate of the condition number of B. It provides a realistic error bound. 😌
  • 'F' — If A = D1 * C * D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the 'C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable. The input matrix A is preprocessed with QR factorization with FULL (row and column) pivoting. 🔥
  • 'G' — Computation as with 'F' with an additional estimate of the condition number of B, where A=B*D. If A has heavily weighted rows, then using this condition number gives too pessimistic error bound. 😭
  • 'A' — Small singular values are not well determined by the data and are considered as noisy; the matrix is treated as numerically rank deficient. The error in the computed singular values is bounded by f(m,n)*epsilon*||A||. The computed SVD A = U * S * V^* restores A up to f(m,n)*epsilon*||A||. This gives the procedure the licence to discard (set to zero) all singular values below N*epsilon*||A||. 💀
  • 'R' — Similar as in 'A'. Rank revealing property of the initial QR factorization is used to reveal (using triangular factor) a gap sigma_{r+1} < epsilon * sigma_r in which case the numerical RANK is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with 'A'. 👑

🦋 JOBU 🦋

JOBU is CHARACTER*1. It specifies whether to compute the columns of U: 💫

  • 'U' — N columns of U are returned in the array U.
  • 'F' — full set of M left sing. vectors is returned in the array U.
  • 'W' — U may be used as workspace of length M*N. See the description of U.
  • 'N' — U is not computed.

🧚 JOBV 🧚

JOBV is CHARACTER*1. It specifies whether to compute the matrix V: 🌈

  • 'V' — N columns of V are returned in the array V; Jacobi rotations are not explicitly accumulated.
  • 'J' — N columns of V are returned in the array V, but they are computed as the product of Jacobi rotations, if JOBT = 'N'.
  • 'W' — V may be used as workspace of length N*N. See the description of V.
  • 'N' — V is not computed.

🌺 JOBR 🌺

JOBR is CHARACTER*1. It specifies the RANGE for the singular values. Issues the licence to set to zero small positive singular values if they are outside specified range. If A .NE. 0 is scaled so that the largest singular value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). 😤

  • 'N' — Do not kill small columns of c*A. This option assumes that BLAS and QR factorizations and triangular solvers are implemented to work in that range. If the condition of A is greater than BIG, use CGESVJ.
  • 'R' — RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)] (roughly, as described above). This option is recommended. ✨

===========================

For computing the singular values in the FULL range [SFMIN,BIG] use CGESVJ. 🔥

🎀 JOBT 🎀

JOBT is CHARACTER*1. If the matrix is square then the procedure may determine to use transposed A if A^* seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. The decision is based on two values of entropy over the adjoint orbit of A^* * A. See the descriptions of RWORK(6) and RWORK(7). 🧠

  • 'T' — transpose if entropy test indicates possibly faster convergence of Jacobi process if A^* is taken as input. If A is replaced with A^*, then the row pivoting is included automatically.
  • 'N' — do not speculate.

The option 'T' can be used to compute only the singular values, or the full SVD (U, SIGMA and V). For only one set of singular vectors (U or V), the caller should provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. The implementer can easily remove this constraint and make the code more complicated. See the descriptions of U and V. In general, this option is considered experimental, and 'N'; should be preferred. This is subject to changes in the future. 🔮

💜 JOBP 💜

JOBP is CHARACTER*1. Issues the licence to introduce structured perturbations to drown denormalized numbers. This licence should be active if the denormals are poorly implemented, causing slow computation, especially in cases of fast convergence (!). For details see [1,2]. For the sake of simplicity, these perturbations are included only when the full SVD or only the singular values are requested. The implementer/user can easily add the perturbation for the cases of computing one set of singular vectors. 🫶

  • 'P' — introduce perturbation
  • 'N' — do not perturb

⭐ 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 COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. 💅

🍓 LDA 🍓

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

🍓 SVA 🍓

SVA is REAL array, dimension (N). On exit:

  • For RWORK(1)/RWORK(2) = ONE: The singular values of A. During the computation SVA contains Euclidean column norms of the iterated matrices in the array A.
  • For RWORK(1) .NE. RWORK(2): The singular values of A are (RWORK(1)/RWORK(2)) * SVA(1:N). This factored form is used if sigma_max(A) overflows or if small singular values have been saved from underflow by scaling the input matrix A.
  • If JOBR='R' then some of the singular values may be returned as exact zeros obtained by … because they are below the numerical rank threshold or are denormalized numbers. 😭

💜 U 💜

(bestie, the original page cuts off right here in part 1 — the rest of U and everything after it is coming in part 2, so don't go anywhere. 🫶)

💖 U 💖

U is a COMPLEX array, dimension ( LDU, N ) or ( LDU, M ).

  • If JOBU = 'U', then on exit U contains the M-by-N matrix of the left singular vectors. slay.
  • If JOBU = 'F', then U contains on exit the M-by-M matrix of the left singular vectors, including an ONB of the orthogonal complement of the Range(A). iconic behavior honestly.
  • If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), then U is used as workspace if the procedure replaces A with A^*. In that case, [V] is computed in U as left singular vectors of A^* and then copied back to the V array. This 'W' option is just a reminder to the caller that in this case U is reserved as workspace of length N*N.
  • If JOBU = 'N', U is not referenced, unless JOBT='T'. bestie it's literally not invited to the function 💅

🌸 LDU 🌸

LDU is INTEGER.

It's the leading dimension of the array U, and LDU >= 1.

IF JOBU = 'U' or 'F' or 'W', then LDU >= M. no cap, check it.

🦋 V 🦋

V is a COMPLEX array, dimension ( LDV, N ).

  • If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of the right singular vectors; yes queen 💅
  • If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N), then V is used as workspace if the procedure replaces A with A^*. In that case, [U] is computed in V as right singular vectors of A^* and then copied back to the U array. This 'W' option is just a reminder to the caller that in this case V is reserved as workspace of length N*N.
  • If JOBV = 'N', V is not referenced, unless JOBT='T'.

🎀 LDV 🎀

LDV is INTEGER.

The leading dimension of the array V, LDV >= 1.

If JOBV = 'V' or 'J' or 'W', then LDV >= N. periodt.

⭐ CWORK ⭐

CWORK is a COMPLEX array, dimension (MAX(2,LWORK)).

If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit CWORK(1) contains the required length of CWORK for the job parameters used in the call. she's telling you what she needs, listen 🫶

👑 LWORK 👑

LWORK is INTEGER.

Length of CWORK to confirm proper allocation of workspace. LWORK depends on the job, and there are so many vibes here, so let's go:

  1. If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
    • LWORK >= 2*N+1. This is the minimal requirement.
    • ->> For optimal performance (blocked code) the optimal value is LWORK >= N + (N+1)*NB. Here NB is the optimal block size for CGEQP3 and CGEQRF.
    • In general, optimal LWORK is computed as LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)). 1.2. .. an estimate of the scaled condition number of A is required (JOBA='E', or 'G'). In this case, LWORK the minimal requirement is LWORK >= N*N + 2*N.
    • ->> For optimal performance (blocked code) the optimal value is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
    • In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ), N*N+LWORK(CPOCON)).
  2. If SIGMA and the right singular vectors are needed (JOBV = 'V'), (JOBU = 'N') 2.1 .. no scaled condition estimate requested (JOBE = 'N'):
    • the minimal requirement is LWORK >= 3*N.
    • For optimal performance, LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for CGEQP3, CGEQRF, CGELQF, CUNMLQ. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ), N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)). 2.2 .. an estimate of the scaled condition number of A is required (JOBA='E', or 'G').
    • the minimal requirement is LWORK >= 3*N.
    • For optimal performance, LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for CGEQP3, CGEQRF, CGELQF, CUNMLQ. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ), N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
  3. If SIGMA and the left singular vectors are needed 3.1 .. no scaled condition estimate requested (JOBE = 'N'):
    • the minimal requirement is LWORK >= 3*N.
    • For optimal performance: if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)). 3.2 .. an estimate of the scaled condition number of A is required (JOBA='E', or 'G').
    • the minimal requirement is LWORK >= 3*N.
    • For optimal performance: if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
  4. If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and 4.1. if JOBV = 'V' the minimal requirement is LWORK >= 5*N+2*N*N. 4.2. if JOBV = 'J' the minimal requirement is LWORK >= 4*N+N*N.

In both cases, the allocated CWORK can accommodate blocked runs of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ. she's flexible like that 💅

If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the minimal length of CWORK for the job parameters used in the call. both options, because she's generous ✨

🌈 RWORK 🌈

RWORK is a REAL array, dimension (MAX(7,LRWORK)).

On exit, bestie, take notes:

  • RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1) such that SCALE*SVA(1:N) are the computed singular values of A. (See the description of SVA().)
  • RWORK(2) = See the description of RWORK(1). yes it's literally just vibing off its sibling 💀
  • RWORK(3) = SCONDA is an estimate for the condition number of column equilibrated A. (If JOBA = 'E' or 'G') SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1). It is computed using CPOCON. It holds N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA where R is the triangular factor from the QRF of A. However, if R is truncated and the numerical rank is determined to be strictly smaller than N, SCONDA is returned as -1, thus indicating that the smallest singular values might be lost. 😭 watch out for that.

If full SVD is needed, the following two condition numbers are useful for the analysis of the algorithm. They are provided for a developer/implementer who is familiar with the details of the method.

  • RWORK(4) = an estimate of the scaled condition number of the triangular factor in the first QR factorization.
  • RWORK(5) = an estimate of the scaled condition number of the triangular factor in the second QR factorization.

The following two parameters are computed if JOBT = 'T'. They are provided for a developer/implementer who is familiar with the details of the method.

  • RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy of diag(A^* * A) / Trace(A^* * A) taken as point in the probability simplex.
  • RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).) twins 🫶

If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit RWORK(1) contains the required length of RWORK for the job parameters used in the call.

🧚 LRWORK 🧚

okay so LRWORK… hold that thought bestie, she's about to eat 💅

  • LRWORK (INTEGER) — the length of RWORK, aka the vibe check that your workspace actually got allocated properly. And bestie, LRWORK depends on the job:
    1. If you're only asking for the singular values, i.e. if LSAME(JOBU,'N') .AND. LSAME(JOBV,'N'), then:
      1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then: LRWORK = max( 7, 2 * M ).
      2. Otherwise, LRWORK = max( 7, N ).
    2. If you want the singular values with the right singular vectors, i.e. if (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND. .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F')), then:
      1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
      2. Otherwise, LRWORK = max( 7, N ).
    3. If you want the singular values with the left singular vectors, i.e. if (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND. .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J')), then:
      1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
      2. Otherwise, LRWORK = max( 7, N ).
    4. If you want the singular values with both the left and the right singular vectors, i.e. if (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND. (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')), then:
      1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
      2. Otherwise, LRWORK = max( 7, N ).

And here's the lil trick 👀 — if, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and the length of RWORK is returned in RWORK(1). Basically you ask "how much room do I need, bestie?" and it tells you.

  • IWORK — an INTEGER array, dimension at least 4, and it further depends on the job, because of course it does:
    1. If only the singular values are requested, then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
    2. If the singular values and the right singular vectors are requested, then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
    3. If the singular values and the left singular vectors are requested, then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
    4. If the singular values with both the left and the right singular vectors are requested, then:
      1. If LSAME(JOBV,'J') the length of IWORK is determined as follows: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
      2. If LSAME(JOBV,'V') the length of IWORK is determined as follows: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

On exit, the receipts are:

  • IWORK(1) = the numerical rank determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR. ✨
  • IWORK(2) = the number of the computed nonzero singular values.
  • IWORK(3) = if nonzero, a warning message: if IWORK(3) = 1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data. 😭
  • IWORK(4) = 1 or -1. If IWORK(4) = 1, then the procedure used A^* to do the job as specified by the JOB parameters.

And if the call to CGEJSV is a workspace query (indicated by LWORK = -1 and LRWORK = -1), then on exit IWORK(1) contains the required length of IWORK for the job parameters used in the call. Iconic.

  • INFO (INTEGER) — the moment of truth, the final verdict:
    • < 0: if INFO = -i, then the i-th argument had an illegal value. Not slay. 🚩
    • = 0: successful exit. We love to see it 💖
    • > 0: CGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.
Heads up bestie: that > 0 case means no convergence, so treat those numbers with suspicion, okay? No cap.

🎀 Author 🎀

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 Further Details: 🌸

CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3, CGEQRF, and CGELQF as preprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor, which in some cases results in much higher accuracy. An example is matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is well-conditioned matrix. In that case, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number of C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if A can be written as A = B*D, with well-conditioned B and some diagonal D, then the high accuracy is guaranteed, both theoretically and in software, independent of D. For more details see [1], [2].

The computational range for the singular values can be the full range ( UNDERFLOW, OVERFLOW ), provided that the machine arithmetic and the BLAS & LAPACK routines called by CGEJSV are implemented to work in that range. If that is not the case, then the restriction for safe computation with the singular values in the range of normalized IEEE numbers is that the spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not overflow. This code (CGEJSV) is best used in this restricted range, meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are returned as zeros. See JOBR for details on this.

Okay so real talk: use it in that restricted range unless your whole stack is built for the full one, fr fr. Anything tiny below ||A||_2 / SLAMCH('O') just comes back as zero. 💀

Further, this implementation is somewhat slower than the one described in [1,2] due to replacement of some non-LAPACK components, and because the choice of some tuning parameters in the iterative part (CGESVJ) is left to the implementer on a particular machine.

The rank revealing QR factorization (in this code: CGEQP3) should be implemented as in [3]. We have a new version of CGEQP3 under development that is more robust than the current one in LAPACK, with a cleaner cut in rank deficient cases. It will be available in the SIGMA library [4]. If M is much larger than N, it is obvious that the initial QRF with column pivoting can be preprocessed by the QRF without pivoting. That well known trick is not used in CGEJSV because in some cases heavy row weighting can be treated with complete pivoting. The overhead in cases M much larger than N is then only due to pivoting, but the benefits in terms of accuracy have prevailed. The implementer/user can incorporate this extra QRF step easily. The implementer can also improve data movement (matrix transpose, matrix copy, matrix transposed copy) — this implementation of CGEJSV uses only the simplest, naive data movement. 🦋

💜 Contributor: 💜

Zlatko Drmac (Zagreb, Croatia)

⭐ References: ⭐

  1. 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.
  2. 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.
  3. Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR factorization software - a case study. ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. LAPACK Working note 176.
  4. Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations. Department of Mathematics, University of Zagreb, 2008, 2016.

🐛 Bugs, examples and comments: 🐛

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you. 🫶

Definition at line 565 of file cgejsv.f.

✨ and now, the main character: DGEJSV ✨

subroutine dgejsv (character*1 joba, character*1 jobu, character*1 jobv,
    character*1 jobr, character*1 jobt, character*1 jobp, integer m, integer
    n, double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( n ) sva, double precision, dimension( ldu, * ) u, integer ldu,
    double precision, dimension( ldv, * ) v, integer ldv, double precision,
    dimension( lwork ) work, integer lwork, integer, dimension( * ) iwork,
    integer info)

DGEJSV

🔥 Purpose: 🔥

DGEJSV computes the singular value decomposition (SVD) of a real M-by-N matrix [A], where M >= N. The SVD of [A] is written as

             [A] = [U] * [SIGMA] * [V]^t,

where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N diagonal elements, [U] is an M-by-N (or M-by-M) 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. The matrices [U] and [V] are computed and stored in the arrays U and V, respectively. The diagonal of [SIGMA] is computed and stored in the array SVA.

DGEJSV can sometimes compute tiny singular values and their singular vectors much more accurately than other SVD routines, see below under Further Details. Obsessed, honestly. 👑

🌈 Parameters 🌈

JOBA

💅 JOBA

okay so JOBA is CHARACTER*1, and it literally sets the accuracy vibe. It specifies the level of accuracy:

  • 'C' — this one slays (high relative accuracy) if A = B * D, with well-conditioned B and arbitrary diagonal matrix D. The accuracy cannot be spoiled by COLUMN scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is bounded by f(M,N)*epsilon* cond(B), independent of D. The input matrix is preprocessed with the QRF with column pivoting. This initial preprocessing and preconditioning by a rank revealing QR factorization is common for all values of JOBA. Additional actions are specified as follows:
  • 'E' — computation as with 'C' with an additional estimate of the condition number of B. It provides a realistic error bound.
  • 'F' — if A = D1 * C * D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the 'C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable. The input matrix A is preprocessed with QR factorization with FULL (row and column) pivoting.
  • 'G' — computation as with 'F' with an additional estimate of the condition number of B, where A=D*B. If A has heavily weighted rows, then using this condition number gives too pessimistic an error bound.
  • 'A' — small singular values are the noise and the matrix is treated as numerically rank deficient. The error in the computed singular values is bounded by f(m,n)*epsilon*||A||. The computed SVD A = U * S * V^t restores A up to f(m,n)*epsilon*||A||. This gives the procedure the licence to discard (set to zero) all singular values below N*epsilon*||A||.
  • 'R' — similar as in 'A'. Rank revealing property of the initial QR factorization is used to reveal (using triangular factor) a gap sigma_{r+1} < epsilon * sigma_r in which case the numerical RANK is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with 'A'.

🫶 JOBU

JOBU is CHARACTER*1. It specifies whether to compute the columns of U:

  • 'U' — N columns of U are returned in the array U.
  • 'F' — full set of M left singular vectors is returned in the array U.
  • 'W' — U may be used as workspace of length M*N. See the description of U.
  • 'N' — U is not computed.

🌸 JOBV

JOBV is CHARACTER*1. It specifies whether to compute the matrix V:

  • 'V' — N columns of V are returned in the array V; Jacobi rotations are not explicitly accumulated.
  • 'J' — N columns of V are returned in the array V, but they are computed as the product of Jacobi rotations. This option is allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
  • 'W' — V may be used as workspace of length N*N. See the description of V.
  • 'N' — V is not computed.

🔥 JOBR

JOBR is CHARACTER*1. It specifies the RANGE for the singular values. It issues the licence to set to zero small positive singular values if they are outside the specified range. If A .NE. 0 is scaled so that the largest singular value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than DSQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').

  • 'N' — do not kill small columns of c*A. This option assumes that BLAS and QR factorizations and triangular solvers are implemented to work in that range. If the condition of A is greater than BIG, use DGESVJ.
  • 'R' — RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)] (roughly, as described above). This option is recommended.
bestie, for computing the singular values in the FULL range [SFMIN,BIG] use DGESVJ.

🦋 JOBT

JOBT is CHARACTER*1. If the matrix is square then the procedure may determine to use transposed A if A^t seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. This is subject to changes in the future. The decision is based on two values of entropy over the adjoint orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).

  • 'T' — transpose if entropy test indicates possibly faster convergence of Jacobi process if A^t is taken as input. If A is replaced with A^t, then the row pivoting is included automatically.
  • 'N' — do not speculate.

This option can be used to compute only the singular values, or the full SVD (U, SIGMA and V). For only one set of singular vectors (U or V), the caller should provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. The implementer can easily remove this constraint and make the code more complicated. See the descriptions of U and V.

🎀 JOBP

JOBP is CHARACTER*1. It issues the licence to introduce structured perturbations to drown denormalized numbers. This licence should be active if the denormals are poorly implemented, causing slow computation, especially in cases of fast convergence (!). For details see [1,2]. For the sake of simplicity, these perturbations are included only when the full SVD or only the singular values are requested. The implementer/user can easily add the perturbation for the cases of computing one set of singular vectors.

  • 'P' — introduce perturbation
  • 'N' — do not perturb

👑 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 DOUBLE PRECISION array, dimension (LDA,N). On entry, the M-by-N matrix A.

💜 LDA

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

🍓 SVA

SVA is DOUBLE PRECISION array, dimension (N). On exit:

  • For WORK(1)/WORK(2) = ONE: the singular values of A. During the computation SVA contains Euclidean column norms of the iterated matrices in the array A.
  • For WORK(1) .NE. WORK(2): the singular values of A are (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if sigma_max(A) overflows or if small singular values have been saved from underflow by scaling the input matrix A.
  • If JOBR='R' then some of the singular values may be returned as exact zeros because they are below the numerical rank threshold or are denormalized numbers.

🧚 U

U is DOUBLE PRECISION array, dimension ( LDU, N ) or ( LDU, M ).

  • If JOBU = 'U', then U contains on exit the M-by-N matrix of the left singular vectors.
  • If JOBU = 'F', then U contains on exit the M-by-M matrix of the left singular vectors, including an ONB of the orthogonal complement of the Range(A).
  • If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), then U is used as workspace if the procedure replaces A with A^t. In that case, [V] is computed in U as left singular vectors of A^t and then copied back to the V array. This 'W' option is just a reminder to the caller that in this case U is reserved as workspace of length N*N.
  • If JOBU = 'N', U is not referenced, unless JOBT='T'.

💖 LDU

LDU is INTEGER. The leading dimension of the array U, LDU >= 1. IF JOBU = 'U' or 'F' or 'W', then LDU >= M.

🧰 V

V is DOUBLE PRECISION array, dimension ( LDV, N ).

  • If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of the right singular vectors;
  • If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N), then V is used as workspace if the procedure replaces A with A^t. In that case, [U] is computed in V as right singular vectors of A^t and then copied back to the U array. This 'W' option is just a reminder to the caller that in this case V is reserved as workspace of length N*N.
  • If JOBV = 'N', V is not referenced, unless JOBT='T'.

🎀 LDV

LDV is INTEGER. The leading dimension of the array V, LDV >= 1. If JOBV = 'V' or 'J' or 'W', then LDV >= N.

🛠️ WORK

okay so we're still in WORK, and here's the rest of the tea: WORK is a DOUBLE PRECISION array, dimension (MAX(7,LWORK)).

On exit, if N > 0 .AND. M > 0 (else not referenced):

  • WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such that SCALE*SVA(1:N) are the computed singular values of A. (See the description of SVA().)
  • WORK(2) = See the description of WORK(1).
  • WORK(3) = SCONDA is an estimate for the condition number of column equilibrated A. (If JOBA = 'E' or 'G') SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). It is computed using DPOCON. It holds N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA where R is the triangular factor from the QRF of A. However, if R is truncated and the numerical rank is determined to be strictly smaller than N, SCONDA is returned as -1, thus indicating that the smallest singular values might be lost.
💀 tiny but important tea: that -1 is literally the "bestie, the smallest singular values might be lost" signal.

If full SVD is needed, the following two condition numbers are useful for the analysis of the algorithm. They are provided for a developer/implementer who is familiar with the details of the method.

  • WORK(4) = an estimate of the scaled condition number of the triangular factor in the first QR factorization.
  • WORK(5) = an estimate of the scaled condition number of the triangular factor in the second QR factorization.

The following two parameters are computed if JOBT = 'T'. They are provided for a developer/implementer who is familiar with the details of the method.

  • WORK(6) = the entropy of A^t*A :: this is the Shannon entropy of diag(A^t*A) / Trace(A^t*A) taken as point in the probability simplex.
  • WORK(7) = the entropy of A*A^t.

💾 LWORK 💾

LWORK is INTEGER Length of WORK to confirm proper allocation of work space. LWORK depends on the job:

  • If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and
    • no scaled condition estimate required (JOBE = 'N'): LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
      • For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal block size for DGEQP3 and DGEQRF. In general, optimal LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
    • an estimate of the scaled condition number of A is required (JOBA='E', 'G'). In this case, LWORK is the maximum of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
      • For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7). In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), N+N*N+LWORK(DPOCON),7).
  • If SIGMA and the right singular vectors are needed (JOBV = 'V'),
    • the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
    • For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF, DORMLQ. In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
  • If SIGMA and the left singular vectors are needed
    • the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
    • For optimal performance:
      • if JOBU = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
      • if JOBU = 'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7), where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR. In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON), 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or M*NB (for JOBU = 'F').
  • If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
    • if JOBV = 'V' the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
    • if JOBV = 'J' the minimal requirement is LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
    • For optimal performance, LWORK should be additionally larger than N+M*NB, where NB is the optimal block size for DORMQR.

🧺 IWORK 🧺

IWORK is INTEGER array, dimension (MAX(3,M+3*N)). On exit,

  • IWORK(1) = the numerical rank determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR.
  • IWORK(2) = the number of the computed nonzero singular values
  • IWORK(3) = if nonzero, a warning message: If IWORK(3) = 1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data.

🚨 INFO 🚨

INFO is INTEGER

  • < 0: if INFO = -i, then the i-th argument had an illegal value.
  • = 0: successful exit;
  • > 0: DGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.

👑 Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🔍 Further Details: 🔍

ngl, DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3, DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor, which in some cases results in much higher accuracy. An example is matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is well-conditioned matrix. In that case, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number of C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if A can be written as A = B*D, with well-conditioned B and some diagonal D, then the high accuracy is guaranteed, both theoretically and in software, independent of D. For more details see [1], [2].

The computational range for the singular values can be the full range ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS & LAPACK routines called by DGEJSV are implemented to work in that range. If that is not the case, then the restriction for safe computation with the singular values in the range of normalized IEEE numbers is that the spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not overflow. This code (DGEJSV) is best used in this restricted range, meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are returned as zeros. See JOBR for details on this.

Further, this implementation is somewhat slower than the one described in [1,2] due to replacement of some non-LAPACK components, and because the choice of some tuning parameters in the iterative part (DGESVJ) is left to the implementer on a particular machine.

The rank revealing QR factorization (in this code: DGEQP3) should be implemented as in [3]. We have a new version of DGEQP3 under development that is more robust than the current one in LAPACK, with a cleaner cut in rank deficient cases. It will be available in the SIGMA library [4].

If M is much larger than N, it is obvious that the initial QRF with column pivoting can be preprocessed by the QRF without pivoting. That well known trick is not used in DGEJSV because in some cases heavy row weighting can be treated with complete pivoting. The overhead in cases M much larger than N is then only due to pivoting, but the benefits in terms of accuracy have prevailed. The implementer/user can incorporate this extra QRF step easily. The implementer can also improve data movement (matrix transpose, matrix copy, matrix transposed copy) - this implementation of DGEJSV uses only the simplest, naive data movement.

🌟 Contributors: 🌟

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

📚 References: 📚

  • [1] 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.
  • [2] 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.
  • [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR factorization software - a case study. ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. LAPACK Working note 176.
  • [4] 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 examples and/or comments to drmac@math.hr. Thank you.

Definition at line 473 of file dgejsv.f.

   subroutine sgejsv (character*1 joba, character*1 jobu, character*1 jobv,
       character*1 jobr, character*1 jobt, character*1 jobp, integer m, integer
       n, real, dimension( lda, * ) a, integer lda, real, dimension( n ) sva,
       real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v,
       integer ldv, real, dimension( lwork ) work, integer lwork, integer,
       dimension( * ) iwork, integer info)

✨ SGEJSV ✨ — okay bestie, this is the one that can sometimes pull tiny singular values and their singular vectors out with way more accuracy than the other SVD routines. Iconic behavior, honestly.

💖 Purpose 💖

SGEJSV computes the singular value decomposition (SVD) of a real M-by-N matrix [A], where M >= N. The SVD of [A] is written as

[A] = [U] * [SIGMA] * [V]^t,

where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N diagonal elements, [U] is an M-by-N (or M-by-M) 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. The matrices [U] and [V] are computed and stored in the arrays U and V, respectively. The diagonal of [SIGMA] is computed and stored in the array SVA.

And yes, SGEJSV can sometimes compute tiny singular values and their singular vectors much more accurately than other SVD routines — see below under Further Details. 🫶

🎀 Parameters 🎀

  • JOBA — CHARACTER*1. This one sets the level of accuracy:
    • 'C' — works well (high relative accuracy) if A = B * D, with well-conditioned B and arbitrary diagonal matrix D. The accuracy cannot be spoiled by COLUMN scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is bounded by f(M,N)*epsilon* cond(B), independent of D. The input matrix is preprocessed with the QRF with column pivoting. This initial preprocessing and preconditioning by a rank revealing QR factorization is common for all values of JOBA. Additional actions are specified as follows:
    • 'E' — computation as with 'C' with an additional estimate of the condition number of B. It provides a realistic error bound.
    • 'F' — if A = D1 * C * D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the 'C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable. The input matrix A is preprocessed with QR factorization with FULL (row and column) pivoting.
    • 'G' — computation as with 'F' with an additional estimate of the condition number of B, where A=D*B. If A has heavily weighted rows, then using this condition number gives too pessimistic error bound.
    • 'A' — small singular values are the noise and the matrix is treated as numerically rank deficient. The error in the computed singular values is bounded by f(m,n)*epsilon*||A||. The computed SVD A = U * S * V^t restores A up to f(m,n)*epsilon*||A||. This gives the procedure the licence to discard (set to zero) all singular values below N*epsilon*||A||.
    • 'R' — similar as in 'A'. Rank revealing property of the initial QR factorization is used do reveal (using triangular factor) a gap sigma_{r+1} < epsilon * sigma_r in which case the numerical RANK is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with 'A'.
  • JOBU — CHARACTER*1. Specifies whether to compute the columns of U:
    • 'U' — N columns of U are returned in the array U.
    • 'F' — full set of M left sing. vectors is returned in the array U.
    • 'W' — U may be used as workspace of length M*N. See the description of U.
    • 'N' — U is not computed.
  • JOBV — CHARACTER*1. Specifies whether to compute the matrix V:
    • 'V' — N columns of V are returned in the array V; Jacobi rotations are not explicitly accumulated.
    • 'J' — N columns of V are returned in the array V, but they are computed as the product of Jacobi rotations. This option is allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
    • 'W' — V may be used as workspace of length N*N. See the description of V.
    • 'N' — V is not computed.
  • JOBR — CHARACTER*1. Specifies the RANGE for the singular values. Issues the licence to set to zero small positive singular values if they are outside specified range. If A .NE. 0 is scaled so that the largest singular value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
    • 'N' — do not kill small columns of c*A. This option assumes that BLAS and QR factorizations and triangular solvers are implemented to work in that range. If the condition of A is greater than BIG, use SGESVJ.
    • 'R' — RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)] (roughly, as described above). This option is recommended.
⚠️ For computing the singular values in the FULL range [SFMIN,BIG] use SGESVJ.
  • JOBT — CHARACTER*1. If the matrix is square then the procedure may determine to use transposed A if A^t seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. This is subject to changes in the future. The decision is based on two values of entropy over the adjoint orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
    • 'T' — transpose if entropy test indicates possibly faster convergence of Jacobi process if A^t is taken as input. If A is replaced with A^t, then the row pivoting is included automatically.
    • 'N' — do not speculate.

This option can be used to compute only the singular values, or the full SVD (U, SIGMA and V). For only one set of singular vectors (U or V), the caller should provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. The implementer can easily remove this constraint and make the code more complicated. See the descriptions of U and V.

  • JOBP — CHARACTER*1. Issues the licence to introduce structured perturbations to drown denormalized numbers. This licence should be active if the denormals are poorly implemented, causing slow computation, especially in cases of fast convergence (!). For details see [1,2]. For the sake of simplicity, this perturbations are included only when the full SVD or only the singular values are requested. The implementer/user can easily add the perturbation for the cases of computing one set of singular vectors.
    • 'P' — introduce perturbation.
    • 'N' — do not perturb.
  • M — INTEGER. The number of rows of the input matrix A. M >= 0.
  • N — INTEGER. The number of columns of the input matrix A. M >= N >= 0.
  • A — REAL array, dimension (LDA,N). On entry, the M-by-N matrix A.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • SVA — REAL array, dimension (N). On exit:
    • For WORK(1)/WORK(2) = ONE: the singular values of A. During the computation SVA contains Euclidean column norms of the iterated matrices in the array A.
    • For WORK(1) .NE. WORK(2): the singular values of A are (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if sigma_max(A) overflows or if small singular values have been saved from underflow by scaling the input matrix A.
    • If JOBR='R' then some of the singular values may be returned as exact zeros obtained by because they are below the numerical rank threshold or are denormalized numbers.
  • U — REAL array, dimension ( LDU, N ) or ( LDU, M ).
    • If JOBU = 'U', then U contains on exit the M-by-N matrix of the left singular vectors.
    • If JOBU = 'F', then U contains on exit the M-by-M matrix of the left singular vectors, including an ONB of the orthogonal complement of the Range(A).
    • If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), then U is used as workspace if the procedure replaces A with A^t. In that case, [V] is computed in U as left singular vectors of A^t and then copied back to the V array. This 'W' option is just a reminder to the caller that in this case U is reserved as workspace of length N*N.
    • If JOBU = 'N' U is not referenced, unless JOBT='T'.
  • LDU

🧵 LDU

LDU is INTEGER, bestie. It's the leading dimension of the array U, and LDU >= 1.

And then — heads up, this is the conditional part — IF JOBU = 'U' or 'F' or 'W', then LDU >= M. No exceptions, that's the vibe check. 💅

🦋 V

V is a REAL array, dimension ( LDV, N ).

What's in it depends on the job, so let's go one by one:

  • If JOBV = 'V', 'J' — then V contains on exit the N-by-N matrix of the right singular vectors. Iconic, as it should.
  • If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N) — then V is used as workspace if the procedure replaces A with A^t. In that case, [U] is computed in V as right singular vectors of A^t and then copied back to the U array. Lowkey the 'W' option is just a little reminder to the caller that in this case V is reserved as workspace of length N*N. That's it, that's the whole point of it.
  • If JOBV = 'N' — V is not referenced, unless JOBT='T'. Sneaky little exception, don't forget it. 🫶

🎀 LDV

LDV is INTEGER — the leading dimension of the array V, and LDV >= 1.

If JOBV = 'V' or 'J' or 'W', then LDV >= N. That's the whole assignment. ✨

🌸 WORK

WORK is a REAL array, dimension (MAX(7,LWORK)). It's doing a lot of heavy lifting on exit, so buckle up:

  • WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such that SCALE*SVA(1:N) are the computed singular values of A. (See the description of SVA().)
  • WORK(2) = See the description of WORK(1). Yes bestie, they point at each other. It's giving mutuals. 💀
  • WORK(3) = SCONDA is an estimate for the condition number of column equilibrated A. (If JOBA = 'E' or 'G'.) SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1). It is computed using SPOCON. It holds:
N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA

where R is the triangular factor from the QRF of A. However — and this is the important caveat, don't skip it — if R is truncated and the numerical rank is determined to be strictly smaller than N, SCONDA is returned as -1, thus indicating that the smallest singular values might be lost. 😭

If full SVD is needed, the following two condition numbers are useful for the analysis of the algorithm. They are provided for a developer/implementer who is familiar with the details of the method:

  • WORK(4) = an estimate of the scaled condition number of the triangular factor in the first QR factorization.
  • WORK(5) = an estimate of the scaled condition number of the triangular factor in the second QR factorization.

The following two parameters are computed if JOBT = 'T'. They are provided for a developer/implementer who is familiar with the details of the method:

  • WORK(6) = the entropy of A^t*A :: this is the Shannon entropy of diag(A^t*A) / Trace(A^t*A) taken as point in the probability simplex.
  • WORK(7) = the entropy of A*A^t.

🔥 LWORK

LWORK is INTEGER — the length of WORK, there to confirm proper allocation of work space. And it depends on the job, so this is the long part. Grab a snack. 🍓

If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and

  • .. no scaled condition estimate required (JOBE = 'N'):
    • LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
      • For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal block size for SGEQP3 and SGEQRF.
      • In general, optimal LWORK is computed as LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SGEQRF), 7).
  • .. an estimate of the scaled condition number of A is required (JOBA='E', 'G'). In this case, LWORK is the maximum of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
    • For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
    • In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SGEQRF), N+N*N+LWORK(SPOCON),7).

If SIGMA and the right singular vectors are needed (JOBV = 'V'),

  • the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
  • For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), where NB is the optimal block size for SGEQP3, SGEQRF, SGELQF, SORMLQ. In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(SGEQP3), N+LWORK(SPOCON), N+LWORK(SGELQF), 2*N+LWORK(SGEQRF), N+LWORK(SORMLQ)).

If SIGMA and the left singular vectors are needed

  • the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
  • For optimal performance:
    • if JOBU = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
    • if JOBU = 'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
    • where NB is the optimal block size for SGEQP3, SGEQRF, SORMQR.
    • In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SPOCON), 2*N+LWORK(SGEQRF), N+LWORK(SORMQR)). Here LWORK(SORMQR) equals N*NB (for JOBU = 'U') or M*NB (for JOBU = 'F').

If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and

  • if JOBV = 'V' the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
  • if JOBV = 'J' the minimal requirement is LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
  • For optimal performance, LWORK should be additionally larger than N+M*NB, where NB is the optimal block size for SORMQR.
bestie please actually read the branch that matches your job args, because picking the wrong LWORK is a whole situation 💀

💜 IWORK

IWORK is INTEGER array, dimension (MAX(3,M+3*N)).

On exit:

  • IWORK(1) = the numerical rank determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR.
  • IWORK(2) = the number of the computed nonzero singular values.
  • IWORK(3) = if nonzero, a warning message: If IWORK(3) = 1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data.

⭐ INFO

INFO is INTEGER. Here's the whole mood ring:

  • < 0: if INFO = -i, then the i-th argument had an illegal value.
  • = 0: successful exit. Slay. 🎉
  • > 0: SGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate. 😭 Not a crash, but definitely not the vibe you were hoping for.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 Further Details:

SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3, SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor, which in some cases results in much higher accuracy. An example is matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is well-conditioned matrix. In that case, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number of C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if A can be written as A = B*D, with well-conditioned B and some diagonal D, then the high accuracy is guaranteed, both theoretically and in software, independent of D. For more details see [1], [2].

The computational range for the singular values can be the full range ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS & LAPACK routines called by SGEJSV are implemented to work in that range. If that is not the case, then the restriction for safe computation with the singular values in the range of normalized IEEE numbers is that the spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not overflow. This code (SGEJSV) is best used in this restricted range, meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are returned as zeros. See JOBR for details on this.

Further, this implementation is somewhat slower than the one described in [1,2] due to replacement of some non-LAPACK components, and because the choice of some tuning parameters in the iterative part (SGESVJ) is left to the implementer on a particular machine.

The rank revealing QR factorization (in this code: SGEQP3) should be implemented as in [3]. We have a new version of SGEQP3 under development that is more robust than the current one in LAPACK, with a cleaner cut in rank deficient cases. It will be available in the SIGMA library [4].

If M is much larger than N, it is obvious that the initial QRF with column pivoting can be preprocessed by the QRF without pivoting. That well known trick is not used in SGEJSV because in some cases heavy row weighting can be treated with complete pivoting. The overhead in cases M much larger than N is then only due to pivoting, but the benefits in terms of accuracy have prevailed. The implementer/user can incorporate this extra QRF step easily. The implementer can also improve data movement (matrix transpose, matrix copy, matrix transposed copy) — this implementation of SGEJSV uses only the simplest, naive data movement.

ngl that last paragraph is basically the authors saying "yeah we know, we left the easy optimizations on the table on purpose" 💅

🌈 Contributors:

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

📚 References:

(keep scrolling bestie, this one's still loading ✨)

💫 References

Not me citing my sources like the academic icon I am 💅 — here's the reading list, exactly as printed:

  1. 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.
  2. 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.
  3. Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR factorization software - a case study. ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. LAPACK Working note 176.
  4. 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

Found a bug? Got a cursed example that haunts you at 3am? Got thoughts? Send literally all of it to drmac@math.hr. They said thank you in advance and they meant it 💖

And yeah — Definition at line 473 of file sgejsv.f. (upstream said it, not me 🌸).

⭐ subroutine zgejsv (…)

Here's the whole signature, no edits, no vibes-based paraphrasing:

subroutine zgejsv (character*1 joba, character*1 jobu, character*1 jobv,
    character*1 jobr, character*1 jobt, character*1 jobp, integer m, integer
    n, complex*16, dimension( lda, * ) a, integer lda, double precision,
    dimension( n ) sva, complex*16, dimension( ldu, * ) u, integer ldu,
    complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension(
    lwork ) cwork, integer lwork, double precision, dimension( lrwork )
    rwork, integer lrwork, integer, dimension( * ) iwork, integer info)
    ZGEJSV

🎀 Purpose

Okay so ZGEJSV computes the singular value decomposition (SVD) of a complex M-by-N matrix [A], where M >= N. And the SVD of [A] is written as

[A] = [U] * [SIGMA] * [V]^*,

where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and [V] is an N-by-N unitary 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. The matrices [U] and [V] are computed and stored in the arrays U and V, respectively. And the diagonal of [SIGMA]? Computed and stored in the array SVA. Iconic division of labour honestly 👑

💖 Parameters

JOBA

JOBA is CHARACTER*1 and it specifies the level of accuracy. This is the main-character option, so pay attention bestie:

  • 'C' — works well (high relative accuracy) if A = B * D, with well-conditioned B and arbitrary diagonal matrix D. Accuracy cannot be spoiled by COLUMN scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is bounded by f(M,N)*epsilon* cond(B), independent of D. The input matrix is preprocessed with the QRF with column pivoting. And psst — this initial preprocessing and preconditioning by a rank revealing QR factorization is common for all values of JOBA. Additional actions get specified below 👇
  • 'E' — computation as with 'C', with an additional estimate of the condition number of B. It provides a realistic error bound. Bestie said "and what's the conditioning, actually?" and honestly fair 🫶
  • 'F' — if A = D1 * C * D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the 'C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable. The input matrix A is preprocessed with QR factorization with FULL (row and column) pivoting. Full pivoting. She's thorough like that 💅
  • 'G' — computation as with 'F' with an additional estimate of the condition number of B, where A=B*D. Heads up though: if A has heavily weighted rows, then using this condition number gives too pessimistic an error bound 🙈
  • 'A' — small singular values are not well determined by the data and are considered as noisy; the matrix is treated as numerically rank deficient. The error in the computed singular values is bounded by f(m,n)*epsilon*||A||. The computed SVD A = U * S * V^* restores A up to f(m,n)*epsilon*||A||. This gives the procedure the licence to discard (set to zero) all singular values below N*epsilon*||A||. She's allowed to throw those away, and she will 😌
  • 'R' — similar as in 'A'. Rank revealing property of the initial QR factorization is used to reveal (using triangular factor) a gap sigma_{r+1} < epsilon * sigma_r, in which case the numerical RANK is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with 'A'. Lowkey the glow-up of 'A' fr fr ✨

JOBU

JOBU is CHARACTER*1 and it specifies whether to compute the columns of U:

  • 'U' — N columns of U are returned in the array U.
  • 'F' — full set of M left sing. vectors is returned in the array U.
  • 'W' — U may be used as workspace of length M*N. See the description of U.
  • 'N' — U is not computed. Respectfully, why 😭 (jk, valid)

JOBV

JOBV is CHARACTER*1 and it specifies whether to compute the matrix V:

  • 'V' — N columns of V are returned in the array V; Jacobi rotations are not explicitly accumulated.
  • 'J' — N columns of V are returned in the array V, but they are computed as the product of Jacobi rotations, if JOBT = 'N'.
  • 'W' — V may be used as workspace of length N*N. See the description of V.
  • 'N' — V is not computed.

JOBR

JOBR is CHARACTER*1 and it specifies the RANGE for the singular values. It issues the licence to set to zero small positive singular values if they are outside the specified range. So: if A .NE. 0 is scaled so that the largest singular value of c*A is around SQRT(BIG), with BIG=DLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN, where SFMIN=DLAMCH('S') and EPSLN=DLAMCH('E').

  • 'N' — do not kill small columns of c*A. This option assumes that BLAS and QR factorizations and triangular solvers are implemented to work in that range. If the condition of A is greater than BIG, use ZGESVJ.
  • 'R' — RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)] (roughly, as described above). This option is recommended. Bestie, take the recommendation 🫶
For computing the singular values in the FULL range [SFMIN,BIG] use ZGESVJ. That's the whole note. Different function, different range, no cap.

JOBT

JOBT is CHARACTER*1. If the matrix is square then the procedure may determine to use transposed A if A^* seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. The decision is based on two values of entropy over the adjoint orbit of A^* * A. See the descriptions of RWORK(6) and RWORK(7) 👀

  • 'T' — transpose if the entropy test indicates possibly faster convergence of the Jacobi process if A^* is taken as input. If A is replaced with A^*, then the row pivoting is included automatically.
  • 'N' — do not speculate. Chill, unbothered, no conjecture 💅

The option 'T' can be used to compute only the singular values, or the full SVD (U, SIGMA and V). For only one set of singular vectors (U or V), the caller should provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. The implementer can easily remove this constraint and make the code more complicated — yes, those are the actual words, I'm not making this up 😭 See the descriptions of U and V.

In general, this option is considered experimental, and 'N' should be preferred. This is subject to changes in the future. Experimental feature, bestie — handle with care 🧚

JOBP

JOBP is CHARACTER*1. It issues the licence to introduce structured perturbations to drown denormalized numbers. This licence should be active if the denormals are poorly implemented, causing slow computation — especially in cases of fast convergence (!). For details see [1,2]. For the sake of simplicity, these perturbations are included only when the full SVD or only the singular values are requested. The implementer/user can easily add the perturbation for the cases of computing one set of singular vectors.

  • 'P' — introduce perturbation. Drown the denormals, they had it coming 🌊
  • 'N' — do not perturb.

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 COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A.

LDA

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

SVA

SVA is DOUBLE PRECISION array, dimension (N). On exit:

  • For RWORK(1)/RWORK(2) = ONE: the singular values of A. During the computation SVA contains Euclidean column norms of the iterated matrices in the array A.
  • For RWORK(1) .NE. RWORK(2): the singular values of A are (RWORK(1)/RWORK(2)) * SVA(1:N). This factored form is used if sigma_max(A) overflows or if small singular values have been saved from underflow by scaling the input matrix A.
  • If JOBR='R' then some of the singular values may be returned as exact zeros — because they are below the numerical rank threshold or are denormalized numbers. (The source text has a lil gap right there, bestie 🫠 but that's the gist.)

U — oh, this one's next, and it's a whole thing… 👀

  • U — COMPLEX*16 array, dimension ( LDU, N ). Okay bestie, here's the whole vibe for U:
    • If JOBU = 'U', then U contains on exit the M-by-N matrix of the left singular vectors. ✨
    • If JOBU = 'F', then U contains on exit the M-by-M matrix of the left singular vectors, including an ONB of the orthogonal complement of the Range(A). 🦋
    • If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N), then U is used as workspace if the procedure replaces A with A^*. In that case, [V] is computed in U as left singular vectors of A^* and then copied back to the V array. This 'W' option is just a reminder to the caller that in this case U is reserved as workspace of length N*N. 💅
    • If JOBU = 'N', U is not referenced, unless JOBT='T'. 🙈
  • LDU — INTEGER. It's the leading dimension of the array U, and LDU >= 1. If JOBU = 'U' or 'F' or 'W', then LDU >= M. That's the rule, no cap. 🎀
  • V — COMPLEX*16 array, dimension ( LDV, N ). Here's the tea on V:
    • If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of the right singular vectors; 🌸
    • If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N), then V is used as workspace if the procedure replaces A with A^*. In that case, [U] is computed in V as right singular vectors of A^* and then copied back to the U array. This 'W' option is just a reminder to the caller that in this case V is reserved as workspace of length N*N. 💖
    • If JOBV = 'N', V is not referenced, unless JOBT='T'. 😌
  • LDV — INTEGER. It's the leading dimension of the array V, and LDV >= 1. If JOBV = 'V' or 'J' or 'W', then LDV >= N. Simple as that. 🫶
  • CWORK — COMPLEX*16 array, dimension (MAX(2,LWORK)). If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit CWORK(1) contains the required length of CWORK for the job parameters used in the call. 👑
  • LWORK — INTEGER. This is the length of CWORK to confirm proper allocation of workspace. LWORK depends on the job, and the breakdown is a whole journey:
  1. If only SIGMA is needed (JOBU = 'N', JOBV = 'N') and
    • 1.1 no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'): LWORK >= 2*N+1. This is the minimal requirement. For optimal performance (blocked code), the optimal value is LWORK >= N + (N+1)*NB. Here NB is the optimal block size for ZGEQP3 and ZGEQRF. In general, optimal LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ)). 🔥
    • 1.2 an estimate of the scaled condition number of A is required (JOBA='E', or 'G'). In this case, the minimal requirement is LWORK >= N*N + 2*N. For optimal performance (blocked code), the optimal value is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ), N*N+LWORK(ZPOCON)). 💜
  2. If SIGMA and the right singular vectors are needed (JOBV = 'V'), (JOBU = 'N')
    • 2.1 no scaled condition estimate requested (JOBE = 'N'): the minimal requirement is LWORK >= 3*N. For optimal performance, LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQF, ZUNMLQ. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZGESVJ), N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)). 🌈
    • 2.2 an estimate of the scaled condition number of A is required (JOBA='E', or 'G'). The minimal requirement is LWORK >= 3*N. For optimal performance, LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQF, ZUNMLQ. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3), LWORK(ZPOCON), N+LWORK(ZGESVJ), N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)). 🧚
  3. If SIGMA and the left singular vectors are needed
    • 3.1 no scaled condition estimate requested (JOBE = 'N'): the minimal requirement is LWORK >= 3*N. For optimal performance: if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)). ⭐
    • 3.2 an estimate of the scaled condition number of A is required (JOBA='E', or 'G'). The minimal requirement is LWORK >= 3*N. For optimal performance: if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB, where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR. In general, the optimal length LWORK is computed as LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)). 🦋
  4. If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
    • 4.1 if JOBV = 'V' the minimal requirement is LWORK >= 5*N+2*N*N. 💀
    • 4.2 if JOBV = 'J' the minimal requirement is LWORK >= 4*N+N*N. 🍓 In both cases, the allocated CWORK can accommodate blocked runs of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, ZUNMLQ. 👑

If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the minimal length of CWORK for the job parameters used in the call. 💖

  • RWORK — DOUBLE PRECISION array, dimension (MAX(7,LRWORK)). On exit, bestie:
    • RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1) such that SCALE*SVA(1:N) are the computed singular values of A. (See the description of SVA().) 🌸
    • RWORK(2) = See the description of RWORK(1).
    • RWORK(3) = SCONDA is an estimate for the condition number of column equilibrated A. (If JOBA = 'E' or 'G') SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1). It is computed using ZPOCON. It holds N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA where R is the triangular factor from the QRF of A. However, if R is truncated and the numerical rank is determined to be strictly smaller than N, SCONDA is returned as -1, thus indicating that the smallest singular values might be lost. 😭

If full SVD is needed, the following two condition numbers are useful for the analysis of the algorithm. They are provided for a developer/implementer who is familiar with the details of the method.

  • RWORK(4) = an estimate of the scaled condition number of the triangular factor in the first QR factorization.
  • RWORK(5) = an estimate of the scaled condition number of the triangular factor in the second QR factorization.

The following two parameters are computed if JOBT = 'T'. They are provided for a developer/implementer who is familiar with the details of the method.

  • RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy of diag(A^* * A) / Trace(A^* * A) taken as point in the probability simplex.
  • RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)

If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or LRWORK=-1), then on exit RWORK(1) contains the required length of RWORK for the job parameters used in the call. 🫶

LRWORK

💾 LRWORK

LRWORK is an INTEGER — it's the length of RWORK, and it's literally here to confirm you allocated enough workspace, bestie. It depends on the job:

  1. If only the singular values are requested, i.e. if LSAME(JOBU,'N') .AND. LSAME(JOBV,'N'), then:
    • 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then: LRWORK = max( 7, 2 * M ).
    • 1.2. Otherwise, LRWORK = max( 7, N ).
  2. If singular values with the right singular vectors are requested, i.e. if (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND. .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F')), then:
    • 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
    • 2.2. Otherwise, LRWORK = max( 7, N ).
  3. If singular values with the left singular vectors are requested, i.e. if (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND. .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J')), then:
    • 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
    • 3.2. Otherwise, LRWORK = max( 7, N ).
  4. If singular values with both the left and the right singular vectors are requested, i.e. if (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND. (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')), then:
    • 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then LRWORK = max( 7, 2 * M ).
    • 4.2. Otherwise, LRWORK = max( 7, N ).

If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and the length of RWORK is returned in RWORK(1).

🧠 IWORK

IWORK is an INTEGER array, of dimension at least 4, that further depends on the job:

  1. If only the singular values are requested then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
  2. If the singular values and the right singular vectors are requested then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
  3. If the singular values and the left singular vectors are requested then: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
  4. If the singular values with both the left and the right singular vectors are requested, then:
    • 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is N+M; otherwise the length of IWORK is N.
    • 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows: if ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

On exit:

  • IWORK(1) = the numerical rank determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR.
  • IWORK(2) = the number of the computed nonzero singular values
  • IWORK(3) = if nonzero, a warning message: If IWORK(3) = 1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data.
  • IWORK(4) = 1 or -1. If IWORK(4) = 1, then the procedure used A^* to do the job as specified by the JOB parameters.

If the call to ZGEJSV is a workspace query (indicated by LWORK = -1 or LRWORK = -1), then on exit IWORK(1) contains the required length of IWORK for the job parameters used in the call.

🚦 INFO

INFO is INTEGER

  • < 0: if INFO = -i, then the i-th argument had an illegal value.
  • = 0: successful exit;
  • > 0: ZGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.

🧑‍💻 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🔍 Further Details

ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3, ZGEQRF, and ZGELQF as preprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor, which in some cases results in much higher accuracy. An example is matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is well-conditioned matrix. In that case, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number of C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if A can be written as A = B*D, with well-conditioned B and some diagonal D, then the high accuracy is guaranteed, both theoretically and in software, independent of D. For more details see [1], [2].

The computational range for the singular values can be the full range ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS & LAPACK routines called by ZGEJSV are implemented to work in that range. If that is not the case, then the restriction for safe computation with the singular values in the range of normalized IEEE numbers is that the spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not overflow. This code (ZGEJSV) is best used in this restricted range, meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are returned as zeros. See JOBR for details on this.

Further, this implementation is somewhat slower than the one described in [1,2] due to replacement of some non-LAPACK components, and because the choice of some tuning parameters in the iterative part (ZGESVJ) is left to the implementer on a particular machine.

The rank revealing QR factorization (in this code: ZGEQP3) should be implemented as in [3]. We have a new version of ZGEQP3 under development that is more robust than the current one in LAPACK, with a cleaner cut in rank deficient cases. It will be available in the SIGMA library [4].

If M is much larger than N, it is obvious that the initial QRF with column pivoting can be preprocessed by the QRF without pivoting. That well known trick is not used in ZGEJSV because in some cases heavy row weighting can be treated with complete pivoting. The overhead in cases M much larger than N is then only due to pivoting, but the benefits in terms of accuracy have prevailed. The implementer/user can incorporate this extra QRF step easily. The implementer can also improve data movement (matrix transpose, matrix copy, matrix transposed copy) - this implementation of ZGEJSV uses only the simplest, naive data movement.

🌟 Contributor

Zlatko Drmac, Department of Mathematics, Faculty of Science, University of Zagreb (Zagreb, Croatia); drmac@math.hr

📚 References

  1. [1] 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.
  2. [2] 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.
  3. [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR factorization software - a case study. ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. LAPACK Working note 176.
  4. [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations. Department of Mathematics, University of Zagreb, 2008, 2016.

🐛 Bugs, examples and comments

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you.

Definition at line 566 of file zgejsv.f.

🤖 Author

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