CLAED7(3)

Welcome to the claed7.f(3) page from the Library Functions Manual 💖 This one is a LAPACK Fortran-routine page, and it is literally giving “dense eigensystem update” main character energy.

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/claed7.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine CLAED7 (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq,
    rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum,
    work, rwork, iwork, info)

CLAED7 used by CSTEDC. 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 CLAED7 💅

subroutine CLAED7 (integer n, integer cutpnt, integer qsiz, integer tlvls,
    integer curlvl, integer curpbm, real, dimension( * ) d, complex,
    dimension( ldq, * ) q, integer ldq, real rho, integer, dimension( * )
    indxq, real, dimension( * ) qstore, integer, dimension( * ) qptr,
    integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer,
    dimension( * ) givptr, integer, dimension( 2, * ) givcol, real,
    dimension( 2, * ) givnum, complex, dimension( * ) work, real, dimension(
    * ) rwork, integer, dimension( * ) iwork, integer info)

CLAED7 used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

🎀 Purpose 🎀

Okay so bestie, CLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form. No cap.

The equation is literally:

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

where Z = Q**Hu, and 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 SLAED2.
2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). 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 🌸

• N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
• CUTPNT — CUTPNT is INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
• QSIZ — QSIZ is INTEGER. The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N.
• 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 REAL array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.
• Q — Q is COMPLEX 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).
• RHO — RHO is REAL. Contains the subdiagonal element used to create the rank-1 modification.
• INDXQ — INDXQ is INTEGER array, dimension (N). This contains the permutation which will reintegrate the subproblem just solved back into sorted order, ie. D( INDXQ( I = 1, N ) ) will be in ascending order.
• IWORK — IWORK is INTEGER array, dimension (4*N).
• RWORK — RWORK is REAL array, dimension (3*N+2*QSIZ*N).
• WORK — WORK is COMPLEX array, dimension (QSIZ*N).
• QSTORE — QSTORE is REAL 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 REAL array, dimension (2, N lg 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. > 0: if INFO = 1, an eigenvalue did not converge.

Lowkey important: those INFO codes are not decorative. = 0 is the slay case, < 0 means an argument was illegal, and > 0 with INFO = 1 means an eigenvalue did not converge. 💀

👑 Author 👑

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

Definition at line 245 of file claed7.f.

🌈 Author 🌈

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