DLAED7(3)

💅 DLAED7(3) — Library Functions Manual 💅

Ok bestie, manual name is folded in, we're in the Library Functions Manual, and this page is about dlaed7. It's giving "LAPACK internal lore, but make it cute." Buckle up. 🦋

💖 NAME 💖

The NAME section is literally just this path, no cap:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaed7.f

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine DLAED7 — signature below, bestie:

subroutine DLAED7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq,
    indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol,
    givnum, work, iwork, info)

DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

🌸 Function/Subroutine Documentation 🌸

subroutine DLAED7 (integer icompq, integer n, integer qsiz, integer tlvls,
    integer curlvl, integer curpbm, double precision, dimension( * ) d,
    double precision, dimension( ldq, * ) q, integer ldq, integer,
    dimension( * ) indxq, double precision rho, integer cutpnt, double
    precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer,
    dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension(
    * ) givptr, integer, dimension( 2, * ) givcol, double precision,
    dimension( 2, * ) givnum, double precision, dimension( * ) work,
    integer, dimension( * ) iwork, integer info)

DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

🎀 Purpose 🎀

So DLAED7 computes the updated eigensystem of a diagonal matrix after a rank-one symmetric matrix comes through and modifies it. This routine is used only for the eigenproblem that requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. DLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired.

The equation is:

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

1. The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED8.
2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED9). This routine also calculates the eigenvectors of the current problem.
3. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

💅 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.
• 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.
• TLVLS — TLVLS is INTEGER. The total number of merging levels in the overall divide and conquer tree.
• CURLVL — CURLVL is INTEGER. The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.
• CURPBM — CURPBM is INTEGER. The current problem in the current level in the overall merge routine (counting from upper left to lower right).
• D — D is DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.
• Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.
• LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N).
• INDXQ — INDXQ is INTEGER array, dimension (N). The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.
• RHO — RHO is DOUBLE PRECISION. The subdiagonal element used to create the rank-1 modification.
• CUTPNT — CUTPNT is INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
• QSTORE — QSTORE is DOUBLE PRECISION array, dimension (N**2+1). Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.
• QPTR — QPTR is INTEGER array, dimension (N+2). List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.
• PRMPTR — PRMPTR is INTEGER array, dimension (N lg N). Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.
• PERM — PERM is INTEGER array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock.
• GIVPTR — GIVPTR is INTEGER array, dimension (N lg N). Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.
• GIVCOL — GIVCOL is INTEGER array, dimension (2, N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
• GIVNUM — GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.
• WORK — WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)
• IWORK — IWORK is INTEGER array, dimension (4*N)
• INFO — INFO is INTEGER
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: if INFO = 1, an eigenvalue did not converge

🚨 Warning-ish vibe check: INFO > 0 with INFO = 1 means an eigenvalue did not converge. That's not a slay. 😭

💜 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 256 of file dlaed7.f. ⭐

Definition at line 256 of file dlaed7.f.

🌈 Author 🌈

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