SRC/sgejsv.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

💖 NAME 💖

SRC/sgejsv.f — from the Library Functions Manual, bestie. This is the SGEJSV page, and it’s lowkey iconic because it’s all about pulling apart a real matrix into its singular-value era. ✨

🌈 SYNOPSIS 🌈

🧩 Functions/Subroutines 🧩

       subroutine sgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda,
           sva, u, ldu, v, ldv, work, lwork, iwork, info)
           SGEJSV

🛠️ Function/Subroutine Documentation 🛠️

   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

✨ Purpose ✨

Okay so 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.

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 🌟

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=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 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. 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 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 🌈

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, 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 🍓

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, 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 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 👑

  • U — a REAL array, dimension ( LDU, N ) or ( LDU, M ). What ends up in it depends on your job vibe:
    • If JOBU = 'U', then U contains on exit the M-by-N matrix of the left singular vectors. cute.
    • 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 Range(A). full send 💅
    • 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. iykyk 🫶
    • If JOBU = 'N', U is not referenced, unless JOBT='T'. respectfully left alone 🙈
  • LDU — INTEGER. The leading dimension of the array U, LDU >= 1. IF JOBU = 'U' or 'F' or 'W', then LDU >= M. no skipping the leading dimension bestie, she matters 🎀
  • V — REAL 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. same energy, mirrored 💫
    • If JOBV = 'N', V is not referenced, unless JOBT='T'.
  • LDV — INTEGER. The leading dimension of the array V, LDV >= 1. If JOBV = 'V' or 'J' or 'W', then LDV >= N. periodt 💜
  • WORK — REAL array, dimension (MAX(7,LWORK)). On exit, she's carrying receipts:
    • 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 really, go look 👀
    • 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, 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. 😭 that -1 is a red flag fr fr

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. (hi devs 👋)

  • 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. iconic math moment 🌈
  • WORK(7) = the entropy of A*A^t. 💀
  • LWORK — INTEGER. Length of WORK to confirm proper allocation of work space. LWORK depends on the job, so listen up:

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. 👑
  • IWORK — 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. 💀 she's telling you the data isn't data-ing
  • INFO — INTEGER
    • < 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 the sweeps giving up

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

(hold that thought bestie, more coming 👇)

like okay so this is the part where SGEJSV spills its whole algorithm-tea ☕✨ — it's a preconditioned Jacobi SVD algorithm, and bestie it is stacked. It calls SGEQP3, SGEQRF, and SGELQF to do the preprocessing and preconditioning work, and optionally you can throw in extra row pivoting as a preprocessor too. And the way that lowkey changes everything?? In some cases it gives you much higher accuracy. 🔥

It's giving… let me paint the picture: take a matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is well-conditioned. In that situation, complete pivoting in the first QR factorizations gives you accuracy that depends on the condition number of C and is independent of D1, D2. Iconic behaviour honestly. 💅 That kind of higher accuracy isn't completely understood theoretically — like, the theory is still catching up — but it works well in practice, fr fr.

And also: if A can be written as A = B*D, with a well-conditioned B and some diagonal D, then high accuracy is guaranteed — both theoretically and in software — independent of D. For more details see [1], [2]. 📚

🌈 the computational range (vibe check)

The computational range for the singular values can be the full range (UNDERFLOW,OVERFLOW) — provided the machine arithmetic and the BLAS & LAPACK routines that SGEJSV calls are implemented to actually work in that range. If they're not, then the restriction for safe computation with 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 singular values of magnitude below ||A||_2 / SLAMCH('O') are returned as zeros. See JOBR for the deets on that. 👀

🐢 okay but why is it kinda slow

Further: this implementation is somewhat slower than the one described in [1,2], and the reasons are lowkey valid — some non-LAPACK components got swapped out, and the choice of some tuning parameters in the iterative part (SGESVJ) is left to whoever implements it on a particular machine. Not me personally, but like. Someone. 🫶

The rank revealing QR factorization (in this code that's SGEQP3) should be implemented as in [3]. Fun fact: there's a new version of SGEQP3 under development that's more robust than the current one in LAPACK, with a cleaner cut in rank deficient cases. It'll be available in the SIGMA library [4]. Periodt. ✨

🌸 if M is much larger than N

Okay so if M is much larger than N, it's obvious that the initial QRF with column pivoting can be preprocessed by the QRF without pivoting. But that well-known trick is not used in SGEJSV, because in some cases heavy row weighting can be treated with complete pivoting. So the overhead in cases where M is much larger than N is then only due to pivoting — but the accuracy benefits have prevailed, and honestly? respect. 👑

The implementer/user can incorporate this extra QRF step easily. And the implementer can also improve the data movement (matrix transpose, matrix copy, matrix transposed copy) — because this implementation of SGEJSV uses only the simplest, naive data movement. No cap. 🧚

🫶 Contributors

  • Zlatko Drmac (Zagreb, Croatia)
  • 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! 🙏💖 (they literally said thank you, so like, be nice and actually send the good examples)

Definition at line 473 of file sgejsv.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

✨ that's the whole page bestie ✨ — courtesy of LAPACK Version 3.12.0, straight out of SRC/sgejsv.f(3). go forth and compute your singular values accurately, ily, mwah 💋🎀💜

📦 where this page comes from