realSYeigen(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 NAME 💖

okay so this is the manual page for realSYeigen(3) from LAPACK, and the NAME section is literally realSYeigen — real. bestie, that's it. No cap.

✨ SYNOPSIS ✨

🦋 Functions 🦋

Here's the whole squad of real symmetric eigenvalue driver routines:

  • subroutine ssyev (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO) — SSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyev_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO) — SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevd (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO) — SSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevd_2stage (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO) — SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevr (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) — SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevr_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) — SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevx (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO) — SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssyevx_2stage (JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO) — SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
  • subroutine ssygv (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO) — SSYGV
  • subroutine ssygv_2stage (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO) — SSYGV_2STAGE
  • subroutine ssygvd (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, IWORK, LIWORK, INFO) — SSYGVD
  • subroutine ssygvx (ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO) — SSYGVX

🌈 Detailed Description 🌈

This is the group of real eigenvalue driver functions for SY matrices. Periodt.

👑 Function Documentation 👑

🌟 subroutine ssyev ... 🌟

Signature time, bestie:

subroutine ssyev (character JOBZ, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real,
    dimension( * ) WORK, integer LWORK, integer INFO)

SSYEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices.

🎯 Purpose 🎯

Okay so SSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.

🎛️ Parameters 🎛️

  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA, N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.
    • If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • W — W is REAL array, dimension (N)
    • If INFO = 0, the eigenvalues in ascending order.
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK))
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER
    • The length of the array WORK. LWORK >= max(1,3*N-1).
    • For optimal efficiency, LWORK >= (NB+2)*N, where NB is the blocksize for SSYTRD returned by ILAENV.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

✍️ Author ✍️

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

🌟 subroutine ssyev_2stage ... 🌟

Now the 2stage version, because we love a glow-up:

subroutine ssyev_2stage (character JOBZ, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real,
    dimension( * ) WORK, integer LWORK, integer INFO)

SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices.

🎯 Purpose 🎯

SSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal.

🎛️ Parameters 🎛️

  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
      • Not available in this release.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA, N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.
    • If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • W — W is REAL array, dimension (N)
    • If INFO = 0, the eigenvalues in ascending order.
  • WORK — WORK is REAL array, dimension LWORK
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER
    • The length of the array WORK. LWORK >= 1, when N <= 1; otherwise
    • If JOBZ = 'N' and N > 1, LWORK must be queried.
    • Then:
LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 2*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.
  • If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

✍️ Author ✍️

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

📚 Further Details 📚

All details about the 2stage techniques are available in:

  • Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
  • A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
  • A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

✨ subroutine ssyevd ✨

subroutine ssyevd (character JOBZ, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real,
    dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
    integer LIWORK, integer INFO)

SSYEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices 🦋

🎀 Purpose 🎀

OK so bestie, SSYEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. And if you're wanting eigenvectors? It goes and uses a divide and conquer algorithm. Iconic behaviour honestly. 💅

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. 🫶

Because of large use of BLAS of level 3, SSYEVD needs N**2 more workspace than SSYEVX. No cap, she's hungry for that WORK array. 😭

🌸 Parameters 🌸

  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA, N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.
    • If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀 bye bye data bestie.
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • W — W is REAL array, dimension (N)
    • If INFO = 0, the eigenvalues in ascending order. Sorted and serving 💅✨
  • WORK — WORK is REAL array, dimension (LWORK)
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER
    • The dimension of the array WORK.
    • If N <= 1, LWORK must be at least 1.
    • If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1.
    • If JOBZ = 'V' and N > 1, LWORK must be at least 1 + 6*N + 2*N**2.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    • On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — LIWORK is INTEGER
    • The dimension of the array IWORK.
    • If N <= 1, LIWORK must be at least 1.
    • If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
    • If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
    • If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
    • if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).

💜 Author 💜

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

🌈 Contributors: 🌈

  • Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
  • Modified by Francoise Tisseur, University of Tennessee
  • Modified description of INFO. Sven, 16 Feb 05.

✨ subroutine ssyevd_2stage ✨

subroutine ssyevd_2stage (character JOBZ, character UPLO, integer N, real,
    dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real,
    dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
    integer LIWORK, integer INFO)

SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices 🦋

🎀 Purpose 🎀

SSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. If eigenvectors are desired, it uses a divide and conquer algorithm. Two stages?? She's doing the most, and honestly? Ate. 👑

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. 🫶

🌸 Parameters 🌸

  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors. Not available in this release. 😭 sorry bestie, it's just not giving yet.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER
    • The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA, N)
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the orthonormal eigenvectors of the matrix A.
    • If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀
  • LDA — LDA is INTEGER
    • The leading dimension of the array A. LDA >= max(1,N).
  • W — W is REAL array, dimension (N)
    • If INFO = 0, the eigenvalues in ascending order. 💅
  • WORK — WORK is REAL array, dimension (LWORK)
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER
    • The dimension of the array WORK.
    • If N <= 1, LWORK must be at least 1.
    • If JOBZ = 'N' and N > 1, LWORK must be queried:
LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 2*N+1
  • where KD is the blocking size of the reduction,
  • FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice
  • NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.
  • If JOBZ = 'V' and N > 1, LWORK must be at least 1 + 6*N + 2*N**2.
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    • On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — LIWORK is INTEGER
    • The dimension of the array IWORK.
    • If N <= 1, LIWORK must be at least 1.
    • If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
    • If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
    • If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • INFO — INFO is INTEGER
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
    • if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).

💜 Author 💜

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

Contributors 🫶

okay so the icons behind this whole thing:

  • Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
  • Modified by Francoise Tisseur, University of Tennessee
  • Modified description of INFO. Sven, 16 Feb 05.

periodt. 💅

Further Details ✨

bestie, if you want all the 2stage tea, it's literally all here:

  • Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
  • A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
  • A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

no cap, these are the receipts. 🔥

subroutine ssyevr 💅

subroutine ssyevr (character JOBZ, character RANGE, character UPLO, integer
    N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer
    IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real,
    dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ,
    real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
    integer LIWORK, integer INFO)

SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

Purpose 🎀

okay so SSYEVR is literally that girl. she computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. you can pick which eigenvalues and eigenvectors you want by specifying either a range of values or a range of indices for the desired eigenvalues. it's giving main character energy, fr fr.

SSYEVR first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows.

For each unreduced block (submatrix) of T,

  • (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
  • (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
  • (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
  • (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see SSTEMR's documentation and:

  • Inderjit S. Dhillon and Beresford N. Parlett: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  • Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
  • Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.
Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made.
Normal execution of SSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Parameters 🍓

okay bestie, here's the full rundown of every parameter. don't skip, iykyk.

  • JOBZ — JOBZ is CHARACTER*1
    • 'N': Compute eigenvalues only;
    • 'V': Compute eigenvalues and eigenvectors.
  • RANGE — RANGE is CHARACTER*1
    • 'A': all eigenvalues will be found.
    • 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • 'I': the IL-th through IU-th eigenvalues will be found. For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and SSTEIN are called
  • UPLO — UPLO is CHARACTER*1
    • 'U': Upper triangle of A is stored;
    • 'L': Lower triangle of A is stored.
  • N — N is INTEGER The order of the matrix A. N >= 0.
  • A — A is REAL array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.
  • LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
  • VL — VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

  • M — M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — W is REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.
  • Z — Z is REAL array, dimension (LDZ, max(1,M)) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe.
  • LDZ — LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • ISUPPZ —
  • ISUPPZ — INTEGER array, dimension ( 2*max(1,M) ). It's the support of the eigenvectors in Z, aka the indices where the nonzero elements of Z actually live. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This one is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by SORMTR. 💫 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1.
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ✨
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,26*N). For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for SSYTRD and SORMTR returned by ILAENV. 🧮 If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. (sneaky little peek, no computation 💅)
  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. 🌸
  • LIWORK — INTEGER. The dimension of the array IWORK. LIWORK >= max(1,10*N). 📏 If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • INFO — INTEGER. The vibe check on the whole operation: 👀
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 😬
    • > 0: Internal error 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 Contributors:

Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USA Jason Riedy, Computer Science Division, University of California at Berkeley, USA

✨ ssyevr_2stage ✨

subroutine ssyevr_2stage (character JOBZ, character RANGE, character UPLO,
    integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU,
    integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W,
    real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * )
    ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( *
    ) IWORK, integer LIWORK, integer INFO)

SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices 👑

🌟 Purpose:

SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 🎯

SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. 🧚

For each unreduced block (submatrix) of T,

  1. Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
  2. Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
  3. For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
  4. For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL. 🎯

For more details, see SSTEMR's documentation and:

  • Inderjit S. Dhillon and Beresford N. Parlett: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
  • Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
  • Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.
Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR_2STAGE calls SSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made.
Heads up bestie 😭 Normal execution of SSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

🌈 Parameters

  • JOBZ — CHARACTER*1. Pick your destiny: ✨
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors. Not available in this release. 😭
  • RANGE — CHARACTER*1. Which eigenvalues we hunting for? 🔍
    • = 'A': all eigenvalues will be found.
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • = 'I': the IL-th through IU-th eigenvalues will be found.
    • For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and SSTEIN are called
  • UPLO — CHARACTER*1. Which triangle are we trusting today? 📐
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — REAL array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. 😌 On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀 no cap, she's gone.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — REAL. The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to 🎯
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. 👑

  • M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — REAL array, dimension (N). The first M elements contain the selected eigenvalues in ascending order. ⬆️✨
  • Z
  • Z — REAL array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then when INFO = 0, the first M columns of Z hold the orthonormal eigenvectors of the matrix A that go with the eigenvalues you selected — the i-th column of Z carries the eigenvector paired with W(i). 💜 If JOBZ = 'N', then Z is not referenced at all, so you can genuinely just not think about it.
⚠️ Note: you have to make sure at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe. 💅
  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N). Periodt.
  • ISUPPZ — INTEGER array, dimension ( 2*max(1,M) ). It's the support of the eigenvectors in Z, i.e. the indices that mark the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by SORMTR. 🦋 Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1.
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ✨
  • LWORK — INTEGER. The dimension of the array WORK.

If JOBZ = 'N' and N > 1, LWORK must be queried:

LWORK = MAX(1, 26*N, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N + 5*N
          = N*KD + N*max(KD+1,FACTOPTNB)
            + max(2*KD*KD, KD*NTHREADS)
            + (KD+1)*N + 5*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 👀

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
  • LIWORK — INTEGER. The dimension of the array IWORK. LIWORK >= max(1,10*N).

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.

  • INFO — INTEGER:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: Internal error 💀

🫶 Author

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

🎀 Contributors:

  • Inderjit Dhillon, IBM Almaden, USA
  • Osni Marques, LBNL/NERSC, USA
  • Ken Stanley, Computer Science Division, University of California at Berkeley, USA
  • Jason Riedy, Computer Science Division, University of California at Berkeley, USA

📚 Further Details:

All the deets about the 2stage techniques live in these, bestie:

  • Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
  • A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
  • A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

🌈 subroutine ssyevx

subroutine ssyevx (character JOBZ, character RANGE, character UPLO, integer
    N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer
    IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real,
    dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer
    LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
    integer INFO)

SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices 🦋

🎯 Purpose:

SSYEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Like, you get to pick. Iconic. 👑

💖 Parameters

  • JOBZ — CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • RANGE — CHARACTER*1
    • = 'A': all eigenvalues will be found.
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO — CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — REAL array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
😭 On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. She's gone. Say bye.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — REAL. The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
        ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). 🙈

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.

  • M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. ⭐
  • Z — REAL array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).

If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 💀

If JOBZ = 'N', then Z is not referenced.

⚠️ Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.
  • LDZ
  • LDZ is INTEGER — the leading dimension of the array Z. So like, LDZ >= 1, and if JOBZ = 'V' then LDZ >= max(1,N). That's the rule bestie, no negotiating 💅
  • WORK is REAL array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. She's got receipts ✨
  • LWORK is INTEGER — the length of the array WORK. LWORK >= 1, when N <= 1; otherwise 8*N. And for optimal efficiency, LWORK >= (NB+3)*N, where NB is the max of the blocksize for SSYTRD and SORMTR returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🫶
  • IWORK is INTEGER array, dimension (5*N)
  • IFAIL is INTEGER array, dimension (N) — if JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced. (so rude of it tbh 😭)
  • INFO is INTEGER
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

👑 Author

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

🦋 subroutine ssyevx_2stage

subroutine ssyevx_2stage (character JOBZ, character RANGE, character UPLO,
    integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU,
    integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W,
    real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK,
    integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * )
    IFAIL, integer INFO)

SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices 🌸

🎀 Purpose

SSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. it's giving main character energy, fr fr 💖

🌈 Parameters

  • JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors. Not available in this release. 😭
  • RANGE is CHARACTER*1
    • = 'A': all eigenvalues will be found.
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N is INTEGER — the order of the matrix A. N >= 0.
  • A is REAL array, dimension (LDA, N) — on entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀 she's gone bestie
  • LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,N).
  • VL is REAL — if RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU is REAL — if RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL is INTEGER — if RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU is INTEGER — if RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL is REAL — the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
        ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3. 📚

  • M is INTEGER — the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W is REAL array, dimension (N) — on normal exit, the first M elements contain the selected eigenvalues in ascending order. ✨ so tidy of them
  • Z is REAL array, dimension (LDZ, max(1,M)) — if JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. 👑 plan ahead bestie
  • LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK is REAL array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK is INTEGER — the length of the array WORK. LWORK >= 1, when N <= 1; otherwise: If JOBZ = 'N' and N > 1, LWORK must be queried.
    LWORK = MAX(1, 8*N, dimension) where
    dimension = max(stage1,stage2) + (KD+1)*N + 3*N
              = N*KD + N*max(KD+1,FACTOPTNB)
                + max(2*KD*KD, KD*NTHREADS)
                + (KD+1)*N + 3*N
    where KD is the blocking size of the reduction,
    FACTOPTNB is the blocking used by the QR or LQ
    algorithm, usually FACTOPTNB=128 is a good choice
    NTHREADS is the number of threads used when
    openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 😭

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

  • IWORK is INTEGER array, dimension (5*N)
  • IFAIL is INTEGER array, dimension (N) — if JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
  • INFO is INTEGER
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

💜 Author

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

🌸 Further Details

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292

like yes bestie, cite your sources, that's academic main character behavior 🎀📚✨

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

subroutine ssygv 💖

   subroutine ssygv (integer ITYPE, character JOBZ, character UPLO, integer N,
       real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
       integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer
       LWORK, integer INFO)

SSYGV — bestie, this is THE glow-up routine for generalized symmetric-definite eigenproblems. ✨

Purpose 🦋

SSYGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric and B is also positive definite. 💅

Parameters 🎀

  • ITYPE — ITYPE is INTEGER. It specifies the problem type to be solved, and honestly it's giving "pick your fighter" energy:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — JOBZ is CHARACTER*1:
    • = 'N': compute eigenvalues only;
    • = 'V': compute eigenvalues and eigenvectors.
  • UPLO — UPLO is CHARACTER*1:
    • = 'U': upper triangles of A and B are stored;
    • = 'L': lower triangles of A and B are stored.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is REAL array, dimension (LDA, N).
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
      • if ITYPE = 1 or 2, Z**T*B*Z = I;
      • if ITYPE = 3, Z**T*inv(B)*Z = I.
    • If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed. 😭 Yeah. Gone. Don't come for her later.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB, N).
    • On entry, the symmetric positive definite matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
    • On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. ⭐
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👑
  • LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= max(1,3*N-1). For optimal efficiency, LWORK >= (NB+2)*N, where NB is the blocksize for SSYTRD returned by ILAENV.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🫶
  • INFO — INFO is INTEGER:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: SPOTRF or SSYEV returned an error code:
      • <= N: if INFO = i, SSYEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
      • > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. 💀

Author 🌸

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

subroutine ssygv_2stage 🔥

   subroutine ssygv_2stage (integer ITYPE, character JOBZ, character UPLO,
       integer N, real, dimension( lda, * ) A, integer LDA, real, dimension(
       ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * )
       WORK, integer LWORK, integer INFO)

SSYGV_2STAGE — same vibe, but ✨two-stage✨, and she's fast on modern hardware. Periodt.

Purpose 🦋

SSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric and B is also positive definite.

This routine use the 2stage technique for the reduction to tridiagonal which showed higher performance on recent architecture and for large sizes N>2000. 🔥

Parameters 🎀

  • ITYPE — ITYPE is INTEGER. It specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — JOBZ is CHARACTER*1:
    • = 'N': compute eigenvalues only;
    • = 'V': compute eigenvalues and eigenvectors. Not available in this release. 🫠
  • UPLO — UPLO is CHARACTER*1:
    • = 'U': upper triangles of A and B are stored;
    • = 'L': lower triangles of A and B are stored.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is REAL array, dimension (LDA, N).
    • On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
    • On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
      • if ITYPE = 1 or 2, Z**T*B*Z = I;
      • if ITYPE = 3, Z**T*inv(B)*Z = I.
    • If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed. 😭 Same energy as before, bestie.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB, N).
    • On entry, the symmetric positive definite matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
    • On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. ⭐
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👑
  • LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise:
If JOBZ = 'N' and N > 1, LWORK must be queried.
                         LWORK = MAX(1, dimension) where
                         dimension = max(stage1,stage2) + (KD+1)*N + 2*N
                                   = N*KD + N*max(KD+1,FACTOPTNB)
                                     + max(2*KD*KD, KD*NTHREADS)
                                     + (KD+1)*N + 2*N
                         where KD is the blocking size of the reduction,
                         FACTOPTNB is the blocking used by the QR or LQ
                         algorithm, usually FACTOPTNB=128 is a good choice
                         NTHREADS is the number of threads used when
                         openMP compilation is enabled, otherwise =1.
  • If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. 💀
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🫶
  • INFO — INFO is INTEGER:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: SPOTRF or SSYEV returned an error code:
      • <= N: if INFO = i, SSYEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
      • > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. 💀

Author 🌸

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

Further Details: 🧚

All details about the 2stage techniques are available in:

  • Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
  • A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
  • A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

🎀 subroutine ssygvd

   subroutine ssygvd (integer ITYPE, character JOBZ, character UPLO, integer N,
       real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
       integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer
       LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
       SSYGVD

✨ Purpose

okay bestie, so SSYGVD computes all the eigenvalues — and if you want them, the eigenvectors too — of a real generalized symmetric-definite eigenproblem. Like, one of these three flavors:

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric and B is also positive definite (non-negotiable, she's gotta be PD 💅). If you want eigenvectors, it uses a divide and conquer algorithm. Iconic behavior, honestly.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. (so like, theoretically a vibe check could fail, but nobody's ever seen it happen 🙈)

🌸 Parameters

  • ITYPE — ITYPE is INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangles of A and B are stored;
    • = 'L': Lower triangles of A and B are stored.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is REAL array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

  • if ITYPE = 1 or 2, Z**T*B*Z = I;
  • if ITYPE = 3, Z**T*inv(B)*Z = I.

If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed. 😭 she's gone, girl.

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB, N). On entry, the symmetric matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.

  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. (yes bestie, sorted, we love an organized queen ✨)
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK.
    • If N <= 1, LWORK >= 1.
    • If JOBZ = 'N' and N > 1, LWORK >= 2*N+1.
    • If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK.
    • If N <= 1, LIWORK >= 1.
    • If JOBZ = 'N' and N > 1, LIWORK >= 1.
    • If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • INFO — INFO is INTEGER
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: SPOTRF or SSYEVD returned an error code:
      • <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1);
      • > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. 💀

👑 Author

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

🦋 Further Details:

Modified so that no backsubstitution is performed if SSYEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

💖 Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

🧚 subroutine ssygvx

   subroutine ssygvx (integer ITYPE, character JOBZ, character RANGE, character
       UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real,
       dimension( ldb, * ) B, integer LDB, real VL, real VU, integer IL,
       integer IU, real ABSTOL, integer M, real, dimension( * ) W, real,
       dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer
       LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
       integer INFO)
       SSYGVX

✨ Purpose

SSYGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. like, you get to pick your faves, no cap 🌈

🌸 Parameters

  • ITYPE — ITYPE is INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • RANGE — RANGE is CHARACTER*1
    • = 'A': all eigenvalues will be found.
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A and B are stored;
    • = 'L': Lower triangle of A and B are stored.
  • N — N is INTEGER. The order of the matrix pencil (A,B). N >= 0.
  • A — A is REAL array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. rest in peace bestie 🪦

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension (LDB, N). On entry, the symmetric matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
  • LDB — INTEGER. The leading dimension of the array B. Gotta be at least LDB >= max(1,N), no skimping bestie.
  • VL — REAL. If RANGE='V', this is the lower bound of the interval we're hunting eigenvalues in, okay? Constraint: VL < VU. Not referenced if RANGE = 'A' or 'I' — so if you're not doing interval mode, she simply doesn't care.
  • VU — REAL. If RANGE='V', this is the upper bound of the interval we're searching for eigenvalues in. Also VL < VU. And again, not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. If RANGE='I', this is the index of the smallest eigenvalue you want back. Rules: 1 <= IL <= IU <= N, if N > 0; and IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. 💅
  • IU — INTEGER. If RANGE='I', this is the index of the largest eigenvalue you want back. Same deal: 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — REAL. The absolute error tolerance for the eigenvalues, aka the vibe threshold for "close enough, we're calling it." An approximate eigenvalue is accepted as converged when it's determined to lie in an interval [a,b] whose width is less than or equal to:
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| gets used in its place instead, where |T| is the 1-norm of the tridiagonal matrix you get by reducing C to tridiagonal form — and C is the symmetric matrix of the standard symmetric problem that the generalized problem gets transformed into. (iykyk 😌)

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. So like, don't just leave it at zero thinking you're being efficient. If this routine comes back with INFO>0, meaning some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').

  • M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1. Math checks out, bestie. ✨
  • W — REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Sorted, organized, thriving. 🎀
  • Z — REAL array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced (she's off duty). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).

The eigenvectors are normalized as follows:

  • if ITYPE = 1 or 2, Z**T*B*Z = I;
  • if ITYPE = 3, Z**T*inv(B)*Z = I.

If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. No cap, plan ahead. 🙈

  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. She's telling you how much room she actually wanted. 🫶
  • LWORK — INTEGER. The length of the array WORK. LWORK >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for SSYTRD returned by ILAENV.

If LWORK = -1, then a workspace query is assumed — the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. It's giving "asking permission first" energy. 💅

  • IWORK — INTEGER array, dimension (5*N).
  • IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
  • INFO — INTEGER. The receipts:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: SPOTRF or SSYEVX returned an error code:
      • <= N: if INFO = i, SSYEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL.
      • > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. 💀

👑 Author 👑

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

💜 Contributors 💜

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

And that's the whole page, bestie — you made it to the end, main character energy fr fr. 🌈 This page is LAPACK's realSYeigen(3), version 3.11.0, dated Sun Nov 27 2022, and it was generated automatically by Doxygen for LAPACK straight from the source code. Iconic. Slay. Periodt. 💖✨🎀

📦 where this page comes from