SRC/dlaed7.f(3)

okay bestie, welcome to SRC/dlaed7.f(3) from the Library Functions Manual — this is the LAPACK page for a dense-matrix eigensystem subroutine, and yes it's giving main-character linear algebra 💅✨

🌸 NAME 🌸

SRC/dlaed7.f

✨ SYNOPSIS ✨

Functions/Subroutines

Bestie, here's the call signature, verbatim, no cap:

       subroutine dlaed7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq,
           indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol,
           givnum, work, iwork, info)

And the little summary that comes with it: 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 💅

Full signature time — again, exactly as written:

   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 🎯

Okay so DLAED7 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 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 formula is literally this:

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 🧾

One bullet per parameter, bestie — all the types, dimensions, and constraints stay exactly as they are:

• ICOMPQ — 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 — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0.
• QSIZ — INTEGER. The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
• TLVLS — INTEGER. The total number of merging levels in the overall divide and conquer tree.
• CURLVL — INTEGER. The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.
• CURPBM — INTEGER. The current problem in the current level in the overall merge routine (counting from upper left to lower right).
• D — DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.
• Q — 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 — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N).
• INDXQ — 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 — DOUBLE PRECISION. The subdiagonal element used to create the rank-1 modification.
• CUTPNT — INTEGER. Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
• QSTORE — 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 — 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 — 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 — INTEGER array, dimension (N lg N). Contains the permutations (from deflation and sorting) to be applied to each eigenblock.
• GIVPTR — 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 — INTEGER array, dimension (2, N lg N). Each pair of numbers indicates a pair of columns to take place in a Givens rotation.
• GIVNUM — DOUBLE PRECISION array, dimension (2, N lg N). Each number indicates the S value to be used in the corresponding Givens rotation.
• WORK — DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N).
• IWORK — INTEGER array, dimension (4*N).
• INFO — 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.

🎓 Author 🎓

Shoutout to the institutions behind this one:

• 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.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

---

💖 that's the whole manual page, bestie. brought to you by LAPACK Version 3.12.0 from SRC/dlaed7.f(3) — mwah 💅✨