SRC/dspgvx.f(3)

🌸 NAME 🌸

SRC/dspgvx.f — that's the page name, bestie, straight from the Library Functions Manual. We're talking about DSPGVX, and it's giving serious linear-algebra main character energy. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine dspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu,
    abstol, m, w, z, ldz, work, iwork, ifail, info)
    DSPGVX

🦋 Function/Subroutine Documentation 🦋

Full signature, no cap:

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 🌈

Okay so 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Iconic behavior, honestly. 💖

🎀 Parameters 🎀

• ITYPE — ITYPE is INTEGER. Specifies the problem type to be solved, and the options are:
  • = 1: A*x = (lambda)*B*x
  • = 2: A*B*x = (lambda)*x
  • = 3: B*A*x = (lambda)*x
• JOBZ — JOBZ is CHARACTER*1. Choose your vibe:
  • = 'N': Compute eigenvalues only;
  • = 'V': Compute eigenvalues and eigenvectors.
• RANGE — RANGE is CHARACTER*1. This is the selection filter, bestie:
  • = '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. Which triangle are we packing?
  • = '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. Not me warning you dramatically, it's literally in the docs.

• 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.

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). 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. 🫶

• 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. Exit-status tea:
  • = 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 principal minor of order i of B is not positive. 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

📌 Definition at line 269 of file dspgvx.f. 📌

💜 Author 💜

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Cute sign-off: Generated automatically by Doxygen for LAPACK from the source code. The source/version/date footer is exactly LAPACK Version 3.12.0 SRC/dspgvx.f(3). Byeeee bestie, go slay those generalized symmetric-definite eigenproblems! 💖✨🎀