doubleOTHEReigen(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 doubleOTHEReigen(3) — LAPACK 💖

✨ NAME ✨

doubleOTHEReigen — double

🦋 SYNOPSIS 🦋

okay so here's the whole squad of functions on this page, bestie:

Functions

  • dbdsvdx — subroutine dbdsvdx (UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, NS, S, Z, LDZ, WORK, IWORK, INFO) — DBDSVDX
  • dggglm — subroutine dggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO) — DGGGLM
  • dsbev — subroutine dsbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO) — DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbev_2stage — subroutine dsbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, INFO) — DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbevd — subroutine dsbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbevd_2stage — subroutine dsbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbevx — subroutine dsbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO) — DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbevx_2stage — subroutine dsbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO) — DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dsbgv — subroutine dsbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO) — DSBGV
  • dsbgvd — subroutine dsbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSBGVD
  • dsbgvx — subroutine dsbgvx (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO) — DSBGVX
  • dspev — subroutine dspev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO) — DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dspevd — subroutine dspevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dspevx — subroutine dspevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO) — DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dspgv — subroutine dspgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO) — DSPGV
  • dspgvd — subroutine dspgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSPGVD
  • dspgvx — subroutine dspgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO) — DSPGVX
  • dstev — subroutine dstev (JOBZ, N, D, E, Z, LDZ, WORK, INFO) — DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dstevd — subroutine dstevd (JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dstevr — subroutine dstevr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) — DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • dstevx — subroutine dstevx (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO) — DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

🌈 Detailed Description 🌈

This is the group of double Other Eigenvalue routines 💅

👑 Function Documentation 👑

🧚 subroutine dbdsvdx 🧚

subroutine dbdsvdx (character UPLO, character JOBZ, character RANGE, integer
    N, double precision, dimension( * ) D, double precision, dimension( * )
    E, double precision VL, double precision VU, integer IL, integer IU,
    integer NS, double precision, dimension( * ) S, double precision,
    dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
    WORK, integer, dimension( * ) IWORK, integer INFO)

DBDSVDX

✨ Purpose ✨

okay bestie, DBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. 💅

Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix

      |  0  d_1                |
      | d_1  0  e_1            |
TGK = |     e_1  0  d_2        |
      |         d_2  .   .     |
      |              .   .   . |

If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].

Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder the values (and corresponding vectors). DBDSVDX implements a) by calling DSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)

💖 Parameters 💖

  • UPLO — UPLO is CHARACTER*1
    • = 'U': B is upper bidiagonal;
    • = 'L': B is lower bidiagonal.
  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute singular values only;
    • = 'V': Compute singular values and singular vectors.
  • RANGE — RANGE is CHARACTER*1
    • = 'A': all singular values will be found.
    • = 'V': all singular values in the half-open interval [VL,VU) will be found.
    • = 'I': the IL-th through IU-th singular values will be found.
  • N — N is INTEGER The order of the bidiagonal matrix. N >= 0.
  • D — D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B.
  • E — E is DOUBLE PRECISION array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1.
  • VL — VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.
  • VU — VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.
  • IL — IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.
  • NS — NS is INTEGER The total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.
  • S — S is DOUBLE PRECISION array, dimension (N) The first NS elements contain the selected singular values in ascending order.
  • Z — Z is DOUBLE PRECISION array, dimension (2*N,K) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i.e.
Z = [ U ]
    [ V ]

If JOBZ = 'N', then Z is not referenced. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

  • LDZ — LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2).
  • WORK — WORK is DOUBLE PRECISION array, dimension (14*N)
  • IWORK — IWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX.
  • INFO — 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 in DSTEVX. The indices of the eigenvectors (as returned by DSTEVX) are stored in the array IWORK.
      • if INFO = N*2 + 1, an internal error occurred.

🌸 Author 🌸

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

🧚 subroutine dggglm 🧚

subroutine dggglm (integer N, integer M, integer P, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, *
    ) B, integer LDB, double precision, dimension( * ) D, double precision,
    dimension( * ) X, double precision, dimension( * ) Y, double precision,
    dimension( * ) WORK, integer LWORK, integer INFO)

DGGGLM

✨ Purpose ✨

DGGGLM solves a general Gauss-Markov linear model (GLM) problem:

        minimize || y ||_2   subject to   d = A*x + B*y
            x

where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. It is assumed that M <= N <= M+P, and

     rank(A) = M    and    rank( A B ) = N.

Under these assumptions, the constrained equation is always consistent 💚, and there's a unique solution x plus a minimal 2-norm solution y, and you get them using a generalized QR factorization of the matrices (A, B), which looks like this bestie:

   A = Q*(R),   B = Q*T*Z.
         (0)

In particular, if the matrix B is square nonsingular, then the problem GLM is literally equivalent to this weighted linear least squares problem:

             minimize || inv(B)*(d-A*x) ||_2
                 x

where inv(B) means the inverse of B. Slay. 💅

✨ Parameters ✨

  • N — N is INTEGER

The number of rows of the matrices A and B. N >= 0.

  • M — M is INTEGER

The number of columns of the matrix A. 0 <= M <= N.

  • P — P is INTEGER

The number of columns of the matrix B. P >= N-M.

  • A — A is DOUBLE PRECISION array, dimension (LDA,M)

On entry, the N-by-M matrix A.

On exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R. 👑

  • LDA — LDA is INTEGER

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

  • B — B is DOUBLE PRECISION array, dimension (LDB,P)

On entry, the N-by-P matrix B.

On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T. 🦋

  • LDB — LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

  • D — D is DOUBLE PRECISION array, dimension (N)

On entry, D is the left hand side of the GLM equation.

On exit, D is destroyed. 💀 rip bestie, it gave everything it had.

  • X — X is DOUBLE PRECISION array, dimension (M)
  • Y — Y is DOUBLE PRECISION array, dimension (P)

On exit, X and Y are the solutions of the GLM problem. ✨

  • WORK — WORK is DOUBLE PRECISION 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. LWORK >= max(1,N+M+P).

For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB is an upper bound for the optimal blocksizes for DGEQRF, SGERQF, DORMQR and SORMRQ.

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. cute, right?

  • INFO — INFO is INTEGER
    • = 0: successful exit. 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • = 1: the upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. 😭
    • = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed. 😭

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsbev

   subroutine dsbev (character JOBZ, character UPLO, integer N, integer KD,
       double precision, dimension( ldab, * ) AB, integer LDAB, double
       precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
       integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

Purpose

DSBEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band 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.

  • KD — KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. 💅

  • LDAB — LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KD + 1.

  • W — W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order. ⭐

  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i).

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

  • LDZ — LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

  • WORK — WORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))
  • 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 dsbev_2stage

   subroutine dsbev_2stage (character JOBZ, character UPLO, integer N, integer
       KD, double precision, dimension( ldab, * ) AB, integer LDAB, double
       precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
       integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
       integer INFO)

DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

Purpose

DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band 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.

  • KD — KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. 💅

  • LDAB — LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KD + 1.

  • W — W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order. ⭐

  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i).

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

  • LDZ — LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

  • WORK — WORK is DOUBLE PRECISION array, dimension LWORK

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  • LWORK

okay bestie we are still in the parameter list, do not scroll away, this is the good part 💅

  • LWORK — LWORK is INTEGER. It's the length of the array WORK. LWORK >= 1, when N <= 1; otherwise 👇
    • If JOBZ = 'N' and N > 1, LWORK must be queried. No guessing, fr fr.
    • LWORK = MAX(1, dimension) where
    • dimension = (2KD+1)*N + KD*NTHREADS + N
    • where KD is the size of the band.
    • 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. 💀 sorry bestie.
    • 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, the receipts 🧾
    • = 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. it just kept going and going and going

💜 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 dsbevd 🌸

subroutine dsbevd (character JOBZ, character UPLO, integer N, integer KD,
    double precision, dimension( ldab, * ) AB, integer LDAB, double
    precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
    integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🦋

Purpose:

DSBEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm.

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. (iconic behaviour, honestly)

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.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. she's not the same matrix she was when she came in 💖

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
  • W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order. ⭐
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. (ghosted, respectfully)
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION 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 > 2, LWORK must be at least 2*N.
    • If JOBZ = 'V' and N > 2, LWORK must be at least ( 1 + 5*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 JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
    • If JOBZ = 'V' and N > 2, 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, 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 dsbevd_2stage 🦋

subroutine dsbevd_2stage (character JOBZ, character UPLO, integer N, integer
    KD, double precision, dimension( ldab, * ) AB, integer LDAB, double
    precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
    integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌈

Purpose:

DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal. If eigenvectors are desired, it uses a divide and conquer algorithm. two stages, zero regrets

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. 😭 she said "not yet" and meant it
  • 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.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.

  • LDAB
  • LDAB — INTEGER energy ✨ this is the leading dimension of the array AB. So like, LDAB >= KD + 1, no exceptions bestie 💅
  • W — DOUBLE PRECISION array, dimension (N). If INFO = 0, this bad boy holds the eigenvalues in ascending order 👑
  • Z — DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then when INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, and the i-th column of Z is holding the eigenvector that goes with W(i) 🫶. If JOBZ = 'N'? Then Z is literally not referenced, she's off the clock 💀
  • LDZ — INTEGER, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N) 🌸
  • WORK — DOUBLE PRECISION array, dimension LWORK. On exit, if INFO = 0, WORK(1) returns the optimal LWORK ⭐
  • LWORK — 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 = (2KD+1)*N + KD*NTHREADS + N
where KD is the size of the band.
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 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 drama 💖

  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK ✨
  • LIWORK — INTEGER, the dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 2, 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. Same vibe, different array 🎀

  • INFO — INTEGER, the receipts 📝
    • = 0: successful exit, we ate 💅
    • < 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 dsbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌈

Purpose: ✨

DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Choose your own adventure, bestie 🎀

Parameters 🌸

  • 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 is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
    • if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    • if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. She changes outfits on the way out 💅

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
  • Q — Q is DOUBLE PRECISION array, dimension (LDQ, N).
    • If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form.
    • If JOBZ = 'N', the array Q is not referenced.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • VL — VL is DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 AB to tridiagonal form.

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

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

  • 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 DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order 👑
  • Z —

Z

Z is a DOUBLE PRECISION array, dimension (LDZ, max(1,M)). okay bestie listen — if JOBZ = 'V', then when INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of matrix A that match the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, that column of Z contains the latest approximation to the eigenvector, and the index of that eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced at all. 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

LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK

WORK is DOUBLE PRECISION array, dimension (7*N).

IWORK

IWORK is INTEGER array, dimension (5*N).

IFAIL

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

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 dsbevx_2stage (...)

subroutine dsbevx_2stage (character JOBZ, character RANGE, character UPLO,
    integer N, integer KD, double precision, dimension( ldab, * ) AB,
    integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ,
    double precision VL, double precision VU, integer IL, integer IU, double
    precision ABSTOL, integer M, double precision, dimension( * ) W, double
    precision, dimension( ldz, * ) Z, integer LDZ, double precision,
    dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
    integer, dimension( * ) IFAIL, integer INFO)

DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. iconic behavior honestly ✨

Purpose

DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band 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.

Parameters

  • JOBZ — JOBZ is CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors. Not available in this release.
  • 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 is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
    • if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    • if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
  • Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • VL — VL is DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 AB to tridiagonal form.

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

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

  • 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 DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order.
  • Z — Z is DOUBLE PRECISION 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. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (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, 7*N, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + 2*N
where KD is the size of the band.
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 — IWORK is INTEGER array, dimension (5*N).
  • IFAIL — 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 — 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
  • 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

🎀 DSBGV 🎀

subroutine dsbgv (character JOBZ, character UPLO, integer N, integer KA,
    integer KB, double precision, dimension( ldab, * ) AB, integer LDAB,
    double precision, dimension( ldbb, * ) BB, integer LDBB, double
    precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
    integer LDZ, double precision, dimension( * ) WORK, integer INFO)

Purpose 💖

Bestie. DSBGV computes all the eigenvalues, and optionally the eigenvectors, of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. That's the whole vibe. ✨

Parameters

  • 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.
  • KA — KA is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
  • KB — KB is INTEGER. The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N).

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed. Gone. Ghosted. 😭
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KA+1.
  • BB — BB is DOUBLE PRECISION array, dimension (LDBB, N).

On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.

  • LDBB — LDBB is INTEGER. The leading dimension of the array BB. LDBB >= KB+1.
  • W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced. (She's off-duty, don't @ her. 💅)
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
  • WORK — WORK is DOUBLE PRECISION array, dimension (3*N)
  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, and i is:
      • <= N: the algorithm 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 DPBSTF returned INFO = i: 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.

🦋 DSBGVD 🦋

subroutine dsbgvd (character JOBZ, character UPLO, integer N, integer KA,
    integer KB, double precision, dimension( ldab, * ) AB, integer LDAB,
    double precision, dimension( ldbb, * ) BB, integer LDBB, double
    precision, dimension( * ) W, double precision, dimension( ldz, * ) Z,
    integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

Purpose 💖

DSBGVD computes all the eigenvalues, and optionally the eigenvectors, of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. Slay. ✨

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.
  • 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.
  • KA — KA is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
  • KB — KB is INTEGER. The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N).

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed. 😭
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KA+1.
  • BB — BB is DOUBLE PRECISION array, dimension (LDBB, N).

On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.

  • LDBB — LDBB is INTEGER. The leading dimension of the array BB. LDBB >= KB+1.
  • W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION 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.
    • If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*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 LIWORK > 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK.
    • If JOBZ = 'N' or 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. so like, it's a vibe check first, no yelling 💅

  • INFO — INFO is INTEGER
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 🙈
    • > 0: if INFO = i, and i is:
      • <= N: the algorithm 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 DPBSTF returned INFO = i: 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

subroutine dsbgvx 👑

subroutine dsbgvx (character JOBZ, character RANGE, character UPLO, integer
    N, integer KA, integer KB, double precision, dimension( ldab, * ) AB,
    integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB,
    double precision, dimension( ldq, * ) Q, integer LDQ, double precision
    VL, double precision VU, integer IL, integer IU, double precision
    ABSTOL, integer M, double precision, dimension( * ) W, double precision,
    dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
    WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
    integer INFO)

DSBGVX

Purpose:

DSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite 🫶. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues — pick your fighter, bestie ✨

Parameters

  • 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 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.

  • KA — KA is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

  • KB — KB is INTEGER

The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.

  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed 💀 like bestie she does not come back the same.
  • LDAB — LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KA+1.

  • BB — BB is DOUBLE PRECISION array, dimension (LDBB, N)

On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.

  • LDBB — LDBB is INTEGER

The leading dimension of the array BB. LDBB >= KB+1.

  • Q — Q is DOUBLE PRECISION array, dimension (LDQ, N)

If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form.

If JOBZ = 'N', the array Q is not referenced 🙅‍♀️

  • LDQ — LDQ is INTEGER

The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

  • VL — VL is DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero ✋. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').

  • 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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order ✨

  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I.

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

  • LDZ — LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

  • WORK — WORK is DOUBLE PRECISION array, dimension (7*N)
  • IWORK — IWORK is INTEGER array, dimension (5*N)
  • IFAIL — IFAIL is INTEGER array, dimension (M)

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 eigenvalues that failed to converge 😭

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

  • INFO — INFO is INTEGER
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL.
    • > N: DPBSTF returned an error code; i.e., 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

subroutine dspev 🦋

subroutine dspev (character JOBZ, character UPLO, integer N, double
    precision, dimension( * ) AP, double precision, dimension( * ) W, double
    precision, dimension( ldz, * ) Z, integer LDZ, double precision,
    dimension( * ) WORK, integer INFO)

DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ✨

Purpose:

DSPEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage 🎀

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.

  • AP — AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A. bestie the input array is gone, it's a canvas now 🎨

  • W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, this is where your eigenvalues are, in ascending order. sorted, bestie, ascending, like a good playlist ✨
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. not referenced = it literally does not care about you 💀
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (3*N). the lil scratchpad, do not perceive it 🫶
  • INFO — INFO is INTEGER.
    • = 0: successful exit. we ate 💅
    • < 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. it tried and it did not slay, sorry

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine dspevd 🦋

subroutine dspevd (character JOBZ, character UPLO, integer N, double
    precision, dimension( * ) AP, double precision, dimension( * ) W, double
    precision, dimension( ldz, * ) Z, integer LDZ, double precision,
    dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
    integer LIWORK, integer INFO)

DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

✨ Purpose ✨

DSPEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. 🌈

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. ngl that's a flex and also a warning, both can be true 😭

🎀 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.
  • AP — AP is DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A. packed AND squished, it's giving minimalism 🎀

  • W — W is DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK. she reports back 🫡
  • 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.
    • If JOBZ = 'V' and N > 1, LWORK must be at least 1 + 6*N + N**2.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the required 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. the -1 trick is iconic, you ask nicely and it just tells you the size, no yelling ✨

  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK.
    • If JOBZ = 'N' or 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 required 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 periodt 💅
    • < 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. not the tridiagonal failing to settle down 😭

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 subroutine dspevx 🌈

subroutine dspevx (character JOBZ, character RANGE, character UPLO, integer
    N, double precision, dimension( * ) AP, double precision VL, double
    precision VU, integer IL, integer IU, double precision ABSTOL, integer
    M, double precision, dimension( * ) W, double precision, dimension( ldz,
    * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer,
    dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

✨ Purpose ✨

DSPEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. you don't have to take all of them bestie, you can be picky 💅

🎀 Parameters 🎀

  • 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 is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • AP — AP is DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A. same deal as before, your array is now a different array, no cap

  • VL — VL is DOUBLE PRECISION. 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 DOUBLE PRECISION. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. yes it's half-open, the VU side is included, the VL side is not, we don't make the rules 🫠
  • 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'. those two are a matched set, don't separate them 🙈
  • ABSTOL — ABSTOL is DOUBLE PRECISION. 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 AP to tridiagonal form. basically: how close is close enough, and if you say "whatever" it picks a sensible default for you 💖

Okay so eigenvalues get computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), and not zero. Like bestie, zero is right there and it's wrong, I'm sorry. And if this routine returns with INFO>0, which is the code for "some eigenvectors did not converge 😭", try setting ABSTOL to 2*DLAMCH('S'). That's the fix. That's it. 🫶

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3. (iykyk, iconic paper, no notes 💅)

  • M — this one is INTEGER. It's the total number of eigenvalues found, and the range is 0 <= M <= N. If RANGE = 'A', then M = N (you asked for all of them, you get all of them, main character energy 👑). And if RANGE = 'I', then M = IU-IL+1.
  • W — W is a DOUBLE PRECISION array, dimension (N). If INFO = 0, it holds the selected eigenvalues in ascending order. So cute, so tidy. ✨
  • Z — Z is a DOUBLE PRECISION array, dimension (LDZ, max(1,M)). Okay so here's the tea: 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. And if JOBZ = 'N', then Z is not referenced at all, she's off duty.
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. So like, over-allocate, bestie, it's not that deep.
  • LDZ — LDZ is INTEGER. It's the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).
  • WORK — WORK is a DOUBLE PRECISION array, dimension (8*N). Eight whole N's, she's hungry. 🍓
  • IWORK — IWORK is an INTEGER array, dimension (5*N).
  • IFAIL — IFAIL is an INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero (all good, all slay 🌸). If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge — like a little receipt of what went wrong. If JOBZ = 'N', then IFAIL is not referenced.
  • INFO — INFO is INTEGER and she is the vibe check:
    • = 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.

💖 DSPGV 💖

   subroutine dspgv (integer ITYPE, character JOBZ, character UPLO, integer N,
       double precision, dimension( * ) AP, double precision, dimension( * )
       BP, double precision, dimension( * ) W, double precision, dimension(
       ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer
       INFO)

🌟 Purpose

DSPGV 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, stored in packed format, and B is also positive definite. (Yes, B has to be the well-behaved one. She's carrying the whole operation. 🫶)

✨ 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.
  • 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.
  • AP — AP is a DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed 💀. Gone. She served her purpose and left, respect.

  • BP — BP is a DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
    • if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    • if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B. So she got a whole glow-up, okay, we love a transformation arc. 🦋

  • W — W is a DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — Z is a DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z 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 Z is not referenced.

  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is a DOUBLE PRECISION array, dimension (3*N).
  • INFO — INFO is INTEGER, the final vibe check:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 💀
    • > 0: DPPTRF or DSPEV returned an error code:
      • <= N: if INFO = i, DSPEV 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. Like, nothing. Zero. The whole thing said no. 😭

👑 Author 👑

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

🌈 DSPGVD 🌈

   subroutine dspgvd (integer ITYPE, character JOBZ, character UPLO, integer N,
       double precision, dimension( * ) AP, double precision, dimension( * )
       BP, double precision, dimension( * ) W, double precision, dimension(
       ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer
       LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

🌟 Purpose

DSPGVD 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, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. ✨ iconic behavior ✨

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, lowkey fine, but we said what we said. 🫶

✨ 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.
  • 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.
  • AP — AP is a DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed 💀.

  • BP — BP is a DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
    • if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    • if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.

  • W — W is a DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — Z is a DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z 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 Z is not referenced.

  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is a DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK. So she tells you what she needed after the fact, which is honestly such a mood. 🫶
  • LWORK — LWORK is INTEGER. The dimension of the array WORK.
    • If N <= 1, LWORK >= 1.
    • If JOBZ = 'N' and N > 1, LWORK >= 2*N.
    • If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

okay so if LWORK = -1, bestie, that's a whole ✨workspace query✨ moment — the routine literally just does the math on how big WORK and IWORK need to be, hands you those sizes as the first entries of the WORK and IWORK arrays, and XERBLA stays completely silent about LWORK or LIWORK. No yelling. No drama. Just vibes and numbers. 💅

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

And if LIWORK = -1, then a workspace query is assumed; the routine only calculates the required 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. (we love a quiet queen 🎀)

  • INFO — INFO is INTEGER, and here's the ✨vibe check✨:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: DPPTRF or DSPEVD returned an error code:
      • <= N: if INFO = i, DSPEVD 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. 💀 literally nothing got computed. devastating.

💖 Author 💖

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

🌸 Contributors 🌸

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

👑 subroutine dspgvx 👑

subroutine dspgvx (integer ITYPE, character JOBZ, character RANGE, character
    UPLO, integer N, double precision, dimension( * ) AP, double precision,
    dimension( * ) BP, double precision VL, double precision VU, integer IL,
    integer IU, double precision ABSTOL, integer M, double precision,
    dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ,
    double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,
    integer, dimension( * ) IFAIL, integer INFO)

DSPGVX ✨

Purpose:

DSPGVX 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, stored in packed storage, and B is also positive definite. (packed storage bestie, we love a compact queen) Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 🦋

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.
  • 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.
  • AP — AP is DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. 💀 gone. reduced to atoms.

  • BP — BP is DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B. 💅

  • VL — VL is DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. 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. (iconic behavior honestly)

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

  • 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 DOUBLE PRECISION array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. ⭐
  • Z — Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced. 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. don't get caught lacking bestie
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (8*N).
  • IWORK — IWORK is INTEGER array, dimension (5*N).
  • IFAIL — 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 — INFO is INTEGER:
    • = 0: successful exit 🎀
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: DPPTRF or DSPEVX returned an error code:
      • <= N: if INFO = i, DSPEVX 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

🦋 subroutine dstev 🦋

subroutine dstev (character JOBZ, integer N, double precision, dimension( * )
    D, double precision, dimension( * ) E, double precision, dimension(
    ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer
    INFO)

DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices (yes, OTHER. we said what we said 💅)

Purpose:

DSTEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. ✨

Parameters

  • JOBZ — JOBZ is CHARACTER*1:
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • N — N is INTEGER. The order of the matrix. N >= 0.
  • D —
  • D — DOUBLE PRECISION array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix A. On exit, if INFO = 0, the eigenvalues in ascending order. 🎀

  • E — DOUBLE PRECISION array, dimension (N-1)

On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to N-1 of E. On exit, bestie… the contents of E are destroyed. 💀 it did its job and then it left, no cap.

  • Z — DOUBLE PRECISION array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with D(i). If JOBZ = 'N', then Z is not referenced — she's off the clock, not even invited. 🙈

  • LDZ — INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 📏

  • WORK — DOUBLE PRECISION array, dimension (max(1,2*N-2))

If JOBZ = 'N', WORK is not referenced. Also just… vibing elsewhere. ✨

  • INFO — INTEGER
    • = 0: successful exit — we ate, no crumbs. 🍓
    • < 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 E did not converge to zero. 😭

💖 Author 💖

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

🦋 subroutine dstevd 🦋

subroutine dstevd (character JOBZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. 💅

✨ Purpose ✨

DSTEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. If eigenvectors are desired, it uses a divide and conquer algorithm. Divide and conquer?? she said let me split this problem and eat it too. 👑

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. 💀 (iconic that they're just like "yeah we don't know of any, moving on")

🌸 Parameters 🌸

  • JOBZ — CHARACTER*1
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • N — INTEGER

The order of the matrix. N >= 0.

  • D — DOUBLE PRECISION array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix A. On exit, if INFO = 0, the eigenvalues in ascending order. 🎀

  • E — DOUBLE PRECISION array, dimension (N-1)

On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to N-1 of E. On exit, the contents of E are destroyed. she served and then she vanished fr fr 💀

  • Z — DOUBLE PRECISION array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with D(i). If JOBZ = 'N', then Z is not referenced. 🙈

  • LDZ — INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 📏

  • WORK — DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🫶

  • LWORK — INTEGER

The dimension of the array WORK. If JOBZ = 'N' or N <= 1 then LWORK must be at least 1. If JOBZ = 'V' and N > 1 then LWORK must be at least ( 1 + 4*N + 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 — INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 🫶

  • LIWORK — INTEGER

The dimension of the array IWORK. If JOBZ = 'N' or N <= 1 then LIWORK must be at least 1. If JOBZ = 'V' and N > 1 then 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 — 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 E did not converge to zero. 😭

👑 Author 👑

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

🦋 subroutine dstevr 🦋

subroutine dstevr (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. 💅

✨ Purpose ✨

DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Pick your fighter. 🎀

Whenever possible, DSTEVR calls DSTEMR to compute the eigenspectrum using Relatively Robust Representations. DSTEMR 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 the i-th unreduced block of T,

  1. Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
  2. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
  3. If there is a cluster of close eigenvalues, 'choose' sigma_i close to the cluster, and go to step (a),
  4. Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.

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

For more details, see 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997. 📚 (yes bestie, they cited the whole thing, respect)

Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and when partial spectrum requests are made.
Normal execution of DSTEMR 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. 💀 no cap, this is the scary part.

🌸 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.
    • For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and DSTEIN are called.
  • N — INTEGER

The order of the matrix. N >= 0.

  • D — DOUBLE PRECISION array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues. 😮‍💨

  • E — DOUBLE PRECISION array, dimension (max(1,N-1))

On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

  • VL — DOUBLE PRECISION

If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.

  • VU — DOUBLE PRECISION

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 — 💅 hold that thought bestie, she's getting her full glow-up in the next part
  • IU — okay so IU is INTEGER. If RANGE='I', it's giving 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'. 💅 periodt
  • ABSTOL — bestie, ABSTOL is DOUBLE PRECISION. It's 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 DLAMCH( '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. ✨ no cap

  • 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 DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order. 🦋
  • Z — Z is DOUBLE PRECISION 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). 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. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 💖
  • ISUPPZ — ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ). The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1. 🧚
  • WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK. 💜
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,20*N).

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. 🔥 fr fr

  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK. ⭐
  • LIWORK — LIWORK is 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 — INFO is 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

🦋 subroutine dstevx 🦋

subroutine dstevx (character JOBZ, character RANGE, integer N, double
    precision, dimension( * ) D, double precision, dimension( * ) E, double
    precision VL, double precision VU, integer IL, integer IU, double
    precision ABSTOL, integer M, double precision, dimension( * ) W, double
    precision, dimension( ldz, * ) Z, integer LDZ, double precision,
    dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension(
    * ) IFAIL, integer INFO)

DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

🎯 Purpose: 🎯

DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

🔥 Parameters 🔥

  • 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.
  • N — N is INTEGER. The order of the matrix. N >= 0.
  • D — D is DOUBLE PRECISION array, dimension (N). On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
  • E — E is DOUBLE PRECISION array, dimension (max(1,N-1)). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
  • VL — VL is DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. 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.

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

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

  • 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 DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order.
  • Z — Z is DOUBLE PRECISION 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 (INFO > 0), 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. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (5*N).
  • IWORK — IWORK is INTEGER array, dimension (5*N).
  • IFAIL
  • IFAIL — INTEGER array, dimension (N) 🫶

Okay so bestie, listen. If JOBZ = 'V', then when INFO = 0, the first M elements of IFAIL are all zero — immaculate, no notes, zero drama. But if INFO > 0? Then IFAIL is holding receipts 📋 — it contains the indices of the eigenvectors that failed to converge. And if JOBZ = 'N', then IFAIL is literally not referenced at all. It's giving "not invited to the function, doesn't even get a mention." 💀

  • INFO — INTEGER ✨

The vibe report, the whole plot, the final verdict:

  • = 0: successful exit 🌈 — we ate, we devoured, nothing burned down
  • < 0: if INFO = -i, the i-th argument had an illegal value 🙈 — someone handed it something cursed
  • > 0: if INFO = i, then i eigenvectors failed to converge 😭 — and their indices are stored in array IFAIL, so you can go find them and ask them what their problem is

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 Author (yes, again, it's a whole thing) 🎀

Generated automatically by Doxygen for LAPACK from the source code. 🤖✨

That's the whole page, bestie — you made it to the end of all 13 parts. Slay. 🫶

And for the official record, since we're not copying the footer line verbatim: this beauty is from LAPACK, Version 3.11.0, dated Sun Nov 27 2022, and it lives at doubleOTHEReigen(3). Icons only. Periodt. 💅🌈

📦 where this page comes from