laed8(3)

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🌸 laed8(3) — Library Functions Manual 🌸

Okay bestie, gather round: this is the manual page for laed8, and the header vibe is literally laed8(3) from the Library Functions Manual. We're folding that manual name in right here so we don't have to copy the top line like a robot 💅.

📛 NAME 📛

laed8 — laed8: D&C step: deflation

✨ SYNOPSIS ✨

🫶 Functions 🫶

  • subroutine claed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info) — CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
  • subroutine dlaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info) — DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
  • subroutine slaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlambda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info) — SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
  • subroutine zlaed8 (k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlambda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info) — ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

💜 Detailed Description 💜

✨ Function Documentation ✨

🧚 subroutine claed8 🧚

subroutine claed8 (integer k, integer n, integer qsiz, complex, dimension(
    ldq, * ) q, integer ldq, real, dimension( * ) d, real rho, integer
    cutpnt, real, dimension( * ) z, real, dimension( * ) dlambda, complex,
    dimension( ldq2, * ) q2, integer ldq2, real, dimension( * ) w, integer,
    dimension( * ) indxp, integer, dimension( * ) indx, integer, dimension(
    * ) indxq, integer, dimension( * ) perm, integer givptr, integer,
    dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, integer info)
    CLAED8 used by CSTEDC. Merges eigenvalues and deflates secular equation.
    Used when the original matrix is dense.

Purpose: 💖

Okay so CLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways deflation can occur: when two or more eigenvalues are close together, or if there is a tiny element in the Z vector. For each such occurrence, the order of the related secular equation problem is reduced by one.

Parameters 🎀

  • K — K is INTEGER. Contains the number of non-deflated eigenvalues. This is the order of the related secular equation.
  • N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
  • QSIZ — QSIZ is INTEGER. The dimension of the unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
  • Q — Q is COMPLEX array, dimension (LDQ,N). On entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max( 1, N ).
  • D — D is REAL array, dimension (N). On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
  • RHO — RHO is REAL. Contains the off diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value required by SLAED3.
  • CUTPNT — CUTPNT is INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. MIN(1,N) <= CUTPNT <= N.
  • Z — Z is REAL array, dimension (N). On input this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). The contents of Z are destroyed during the updating process.
  • DLAMBDA — DLAMBDA is REAL array, dimension (N). Contains a copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation.
  • Q2 — Q2 is COMPLEX array, dimension (LDQ2,N). If ICOMPQ = 0, Q2 is not referenced. Otherwise, contains a copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (SGEMM) to update the new eigenvectors.
  • LDQ2 — LDQ2 is INTEGER. The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
  • W — W is REAL array, dimension (N). This will hold the first k values of the final deflation-altered z-vector and will be passed to SLAED3.
  • INDXP — INDXP is INTEGER array, dimension (N). This will contain the permutation used to place deflated values of D at the end of the array. On output INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.
  • INDX — INDX is INTEGER array, dimension (N). This will contain the permutation used to sort the contents of D into ascending order.
  • INDXQ — INDXQ is INTEGER array, dimension (N). This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate.
  • PERM — PERM is INTEGER array, dimension (N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock.
  • GIVPTR — GIVPTR is INTEGER. Contains the number of Givens rotations which took place in this subproblem.
  • GIVCOL — GIVCOL is INTEGER array, dimension (2, N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
  • GIVNUM — GIVNUM is REAL array, dimension (2, N). Each number indicates the S value to be used in the corresponding Givens rotation.
  • INFO — INFO is INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.

Author 👑

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

Definition at line 225 of file claed8.f.

🦋 subroutine dlaed8 🦋

subroutine dlaed8 (integer icompq, integer k, integer n, integer qsiz,
    double precision, dimension( * ) d, double precision, dimension( ldq, *
    ) q, integer ldq, integer, dimension( * ) indxq, double precision rho,
    integer cutpnt, double precision, dimension( * ) z, double precision,
    dimension( * ) dlambda, double precision, dimension( ldq2, * ) q2,
    integer ldq2, double precision, dimension( * ) w, integer, dimension( *
    ) perm, integer givptr, integer, dimension( 2, * ) givcol, double
    precision, dimension( 2, * ) givnum, integer, dimension( * ) indxp,
    integer, dimension( * ) indx, integer info)
    DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation.
    Used when the original matrix is dense.

Purpose: 💖

DLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways deflation can occur: when two or more eigenvalues are close together, or if there is a tiny element in the Z vector. For each such occurrence, the order of the related secular equation problem is reduced by one.

Parameters 🎀

  • ICOMPQ — ICOMPQ is INTEGER.
    • = 0: Compute eigenvalues only.
    • = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.
  • K — K is INTEGER. The number of non-deflated eigenvalues, and the order of the related secular equation.
  • N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
  • QSIZ — QSIZ is INTEGER. The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
  • D — D is DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
  • Q — Q is DOUBLE PRECISION array, dimension (LDQ,N). If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N).
  • INDXQ —

okay so picking right back up with the rest of the dlaed8 params, bestie 💖 — same energy, all the facts:

  • INDXQ — INTEGER array, dimension (N). This is the permutation that separately sorts the two sub-problems in D into ascending order. ⚠️ Heads up though: elements in the second half of this permutation have to get CUTPNT added to their values first before they're accurate. Like, don't skip that step, fr fr.
  • RHO — DOUBLE PRECISION. On entry, it's the off-diagonal element tied to the rank-1 cut that originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3. ✨ glow-up on exit ✨
  • CUTPNT — INTEGER. The location of the last eigenvalue in the leading sub-matrix. It's range-checked as min(1,N) <= CUTPNT <= N. 🎯
  • Z — DOUBLE PRECISION array, dimension (N). On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process 😭 like it literally does not survive, rest in peace.
  • DLAMBDA — DOUBLE PRECISION array, dimension (N). A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation. 🎀
  • Q2 — DOUBLE PRECISION array, dimension (LDQ2,N). If ICOMPQ = 0, Q2 is not referenced. Otherwise, it's a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors. 🦋
  • LDQ2 — INTEGER. The leading dimension of the array Q2. Constraint: LDQ2 >= max(1,N). Don't test it. 💅
  • W — DOUBLE PRECISION array, dimension (N). The first k values of the final deflation-altered z-vector, and will be passed to DLAED3. ✨
  • PERM — INTEGER array, dimension (N). The permutations (from deflation and sorting) to be applied to each eigenblock. 🔀
  • GIVPTR — INTEGER. The number of Givens rotations which took place in this subproblem. Count 'em 👑
  • GIVCOL — INTEGER array, dimension (2, N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
  • GIVNUM — DOUBLE PRECISION array, dimension (2, N). Each number indicates the S value to be used in the corresponding Givens rotation. ⭐
  • INDXP — INTEGER array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues. 🍓
  • INDX — INTEGER array, dimension (N). The permutation used to sort the contents of D into ascending order. 📈
  • INFO — INTEGER. Exit status, and it's giving very simple:
    • = 0: successful exit. slay 😌
    • < 0: if INFO = -i, the i-th argument had an illegal value. 💀

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🫶 Contributors: 🫶

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file dlaed8.f.

🦋 subroutine slaed8 🦋

okay so now we vibe-switch to the single-precision twin, and here's the whole signature verbatim so you can screenshot it:

subroutine slaed8 (integer icompq, integer k, integer n, integer qsiz, real,
    dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer,
    dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) z,
    real, dimension( * ) dlambda, real, dimension( ldq2, * ) q2, integer
    ldq2, real, dimension( * ) w, integer, dimension( * ) perm, integer
    givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * )
    givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx,
    integer info)

SLAED8 is used by SSTEDC. It merges eigenvalues and deflates the secular equation. Used when the original matrix is dense. 🔥

✨ Purpose: ✨

SLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. 🎀

💜 Parameters 💜

  • ICOMPQ — INTEGER. Mode switch, pick your fighter:
    • = 0: Compute eigenvalues only.
    • = 1: Compute eigenvectors of the original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. 👑
  • K — INTEGER. The number of non-deflated eigenvalues, and the order of the related secular equation. 🔢
  • N — INTEGER. The dimension of the symmetric tridiagonal matrix. Constraint: N >= 0. 🌈
  • QSIZ — INTEGER. The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. Constraint: QSIZ >= N if ICOMPQ = 1. 🦋
  • D — REAL array, dimension (N). On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order. 📊
  • Q — REAL array, dimension (LDQ,N). If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns. 💫
  • LDQ — INTEGER. The leading dimension of the array Q. Constraint: LDQ >= max(1,N). 💅
  • INDXQ — INTEGER array, dimension (N). The permutation which separately sorts the two sub-problems in D into ascending order. Again, the note that matters: elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate. ⚠️ do not forget, bestie.
  • RHO — REAL. On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by SLAED3. ✨
  • CUTPNT — INTEGER. The location of the last eigenvalue in the leading sub-matrix. Range: min(1,N) <= CUTPNT <= N. 🎯
  • Z — REAL array, dimension (N). On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process 😭 gone, reduced to atoms.
  • DLAMBDA — REAL array, dimension (N). A copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation. 🎀
  • Q2 — REAL array, dimension (LDQ2,N). If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (SGEMM) to update the new eigenvectors. 🌸
  • LDQ2 — INTEGER. The leading dimension of the array Q2. Constraint: LDQ2 >= max(1,N). 🫶
  • W — REAL array, dimension (N). The first k values of the final deflation-altered z-vector and will be passed to SLAED3. ⭐
  • PERM — INTEGER array, dimension (N). The permutations (from deflation and sorting) to be applied to each eigenblock. 🔀
  • GIVPTR — INTEGER. The number of Givens rotations which took place in this subproblem. 👑
  • GIVCOL — INTEGER array, dimension (2, N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
  • GIVNUM — REAL array, dimension (2, N). Each number indicates the S value to be used in the corresponding Givens rotation. 💖
  • INDXP — INTEGER array, dimension (N). The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues. 🍓
  • INDX — INTEGER array, dimension (N). The permutation used to sort the contents of D into ascending order. 📈
  • INFO — INTEGER. Exit status, same deal as always:
    • = 0: successful exit. 😌
    • < 0: if INFO = -i, the i-th argument had an illegal value. 💀

🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 Contributors: 🧚

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file slaed8.f.

💖 Parameters 💖

okay bestie, this is the full parameter roll call — every single one, no skips, no cap ✨

  • K — INTEGER. Contains the number of non-deflated eigenvalues. This is the order of the related secular equation. 💅
  • N — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
  • QSIZ — INTEGER. The dimension of the unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
  • Q — COMPLEX*16 array, dimension (LDQ,N). On entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns. 🦋
  • LDQ — INTEGER. The leading dimension of the array Q. LDQ >= max( 1, N ).
  • D — DOUBLE PRECISION array, dimension (N). On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
  • RHO — DOUBLE PRECISION. Contains the off diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value required by DLAED3.
  • CUTPNT — INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. MIN(1,N) <= CUTPNT <= N.
  • Z — DOUBLE PRECISION array, dimension (N). On input this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). The contents of Z are destroyed during the updating process. 😭
  • DLAMBDA — DOUBLE PRECISION array, dimension (N). Contains a copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.
  • Q2 — COMPLEX*16 array, dimension (LDQ2,N). If ICOMPQ = 0, Q2 is not referenced. Otherwise, contains a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors. 🔥
  • LDQ2 — INTEGER. The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
  • W — DOUBLE PRECISION array, dimension (N). This will hold the first k values of the final deflation-altered z-vector and will be passed to DLAED3.
  • INDXP — INTEGER array, dimension (N). This will contain the permutation used to place deflated values of D at the end of the array. On output INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues. 🎀
  • INDX — INTEGER array, dimension (N). This will contain the permutation used to sort the contents of D into ascending order.
  • INDXQ — INTEGER array, dimension (N). This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate.
  • PERM — INTEGER array, dimension (N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock.
  • GIVPTR — INTEGER. Contains the number of Givens rotations which took place in this subproblem.
  • GIVCOL — INTEGER array, dimension (2, N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
  • GIVNUM — DOUBLE PRECISION array, dimension (2, N). Each number indicates the S value to be used in the corresponding Givens rotation. 🌸
  • INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 💀

🌸 Author 🌸

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

Definition at line 225 of file zlaed8.f.

✨ Author ✨

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