laed0(3)

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version 3.12.1 · 9 versions
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💅 laed0(3) — Library Functions Manual, bestie edition ✨

okay so this is laed0 from section 3 of the Library Functions Manual, and buckle up because it's a whole family of subroutines: claed0, dlaed0, slaed0, zlaed0. They all do the same iconic thing in different precisions/flavours. The name laed0 lowkey means "D&C step: top level solver" — divide and conquer, main character energy. 🌈

✨ NAME ✨

laed0 — laed0: D&C step: top level solver 🎀

🦋 SYNOPSIS 🦋

Functions

  • subroutine claed0 (qsiz, n, d, e, q, ldq, qstore, ldqs, rwork, iwork, info) — CLAED0 used by CSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 💜
  • subroutine dlaed0 (icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info) — DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 💜
  • subroutine slaed0 (icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info) — SLAED0 used by SSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 💜
  • subroutine zlaed0 (qsiz, n, d, e, q, ldq, qstore, ldqs, rwork, iwork, info) — ZLAED0 used by ZSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 💜

🌸 Detailed Description 🌸

ngl, the detailed description section is just vibes — the real tea is all in Function Documentation below. 🫶

👑 Function Documentation 👑

🌟 subroutine claed0 🌟

subroutine claed0 (integer qsiz, integer n, real, dimension( * ) d, real,
    dimension( * ) e, complex, dimension( ldq, * ) q, integer ldq, complex,
    dimension( ldqs, * ) qstore, integer ldqs, real, dimension( * ) rwork,
    integer, dimension( * ) iwork, integer info)

CLAED0 used by CSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. ✨

Purpose:

Using the divide and conquer method, CLAED0 computes all eigenvalues of a symmetric tridiagonal matrix which is one diagonal block of those from reducing a dense or band Hermitian matrix and corresponding eigenvectors of the dense or band matrix.

Parameters

  • QSIZ — QSIZ is INTEGER. The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. 🎀
  • N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0. ✨
  • D — D is REAL array, dimension (N). On entry, the diagonal elements of the tridiagonal matrix. On exit, the eigenvalues in ascending order. 💖
  • E — E is REAL array, dimension (N-1). On entry, the off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed. 💀 (rip, she's gone, that's canon)
  • Q — Q is COMPLEX array, dimension (LDQ,N). On entry, Q must contain an QSIZ x N matrix whose columns unitarily orthonormal. It is a part of the unitary matrix that reduces the full dense Hermitian matrix to a (reducible) symmetric tridiagonal matrix. 🦋
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N). ⭐
  • IWORK — IWORK is INTEGER array, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N
( lg( N ) = smallest integer k
            such that 2^k >= N )

🧚

  • RWORK — RWORK is REAL array, dimension (1 + 3*N + 2*N*lg N + 3*N**2)
( lg( N ) = smallest integer k
            such that 2^k >= N )

🧚

  • QSTORE — QSTORE is COMPLEX array, dimension (LDQS, N). Used to store parts of the eigenvector matrix when the updating matrix multiplies take place. 🌸
  • LDQS — LDQS is INTEGER. The leading dimension of the array QSTORE. LDQS >= max(1,N). ⭐
  • INFO — INFO is INTEGER. 🎀
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).

Author

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

Definition at line 143 of file claed0.f. 📍

🌟 subroutine dlaed0 🌟

subroutine dlaed0 (integer icompq, integer qsiz, integer n, double
    precision, dimension( * ) d, double precision, dimension( * ) e, double
    precision, dimension( ldq, * ) q, integer ldq, double precision,
    dimension( ldqs, * ) qstore, integer ldqs, double precision, dimension(
    * ) work, integer, dimension( * ) iwork, integer info)

DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 🐬

Purpose:

DLAED0 computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method.

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.
    • = 2: Compute eigenvalues and eigenvectors of tridiagonal matrix.
  • QSIZ — QSIZ is INTEGER. The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. 🎀
  • N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0. ✨
  • D — D is DOUBLE PRECISION array, dimension (N). On entry, the main diagonal of the tridiagonal matrix. On exit, its eigenvalues. 💖
  • E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed. 💀
  • Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). On entry, Q must contain an N-by-N orthogonal matrix. 🌈
    • If ICOMPQ = 0 — Q is not referenced.
    • If ICOMPQ = 1 — On entry, Q is a subset of the columns of the orthogonal matrix used to reduce the full matrix to tridiagonal form corresponding to the subset of the full matrix which is being decomposed at this time.
    • If ICOMPQ = 2 — On entry, Q will be the identity matrix. On exit, Q contains the eigenvectors of the tridiagonal matrix.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. If eigenvectors are desired, then LDQ >= max(1,N). In any case, LDQ >= 1. ⭐
  • QSTORE — QSTORE is DOUBLE PRECISION array, dimension (LDQS, N). Referenced only when ICOMPQ = 1. Used to store parts of the eigenvector matrix when the updating matrix multiplies take place. 🌸
  • LDQS — LDQS is INTEGER. The leading dimension of the array QSTORE. If ICOMPQ = 1, then LDQS >= max(1,N). In any case, LDQS >= 1. ⭐
  • WORK — WORK is DOUBLE PRECISION array. 🧚
    • If ICOMPQ = 0 or 1, the dimension of WORK must be at least
1 + 3*N + 2*N*lg N + 3*N**2
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • If ICOMPQ = 2, the dimension of WORK must be at least
4*N + N**2.
  • IWORK — IWORK is INTEGER array. 🧚
    • If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N.
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • If ICOMPQ = 2, the dimension of IWORK must be at least
3 + 5*N.
  • INFO — INFO is INTEGER. 🎀
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).

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 170 of file dlaed0.f. 📍

🌟 subroutine slaed0 🌟

subroutine slaed0 (integer icompq, integer qsiz, integer n, real, dimension(
    * ) d, real, dimension( * ) e, real, dimension( ldq, * ) q, integer ldq,
    real, dimension( ldqs, * ) qstore, integer ldqs, real, dimension( * )
    work, integer, dimension( * ) iwork, integer info)

SLAED0 used by SSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. 🔥

Purpose:

SLAED0 computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method.

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.
    • = 2: Compute eigenvalues and eigenvectors of tridiagonal matrix.
  • QSIZ —

continuing the parameter list — we're still in SLAED0 land, bestie 💅

  • QSIZ — INTEGER. It's the dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. Also: QSIZ >= N if ICOMPQ = 1 👑
  • N — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0 ✨
  • D — REAL array, dimension (N). On entry, it's the main diagonal of the tridiagonal matrix. On exit, it's holding its eigenvalues 🎀
  • E — REAL array, dimension (N-1). These are the off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed 💀
  • Q — REAL array, dimension (LDQ, N). On entry, Q must contain an N-by-N orthogonal matrix.
    • If ICOMPQ = 0: Q is not referenced 🙈
    • If ICOMPQ = 1: on entry, Q is a subset of the columns of the orthogonal matrix used to reduce the full matrix to tridiagonal form, corresponding to the subset of the full matrix which is being decomposed at this time 🦋
    • If ICOMPQ = 2: on entry, Q will be the identity matrix. On exit, Q contains the eigenvectors of the tridiagonal matrix 👑
  • LDQ — INTEGER. The leading dimension of the array Q. If eigenvectors are desired, then LDQ >= max(1,N). In any case, LDQ >= 1 🌸
  • QSTORE — REAL array, dimension (LDQS, N). Referenced only when ICOMPQ = 1. Used to store parts of the eigenvector matrix when the updating matrix multiplies take place 🫶
  • LDQS — INTEGER. The leading dimension of the array QSTORE. If ICOMPQ = 1, then LDQS >= max(1,N). In any case, LDQS >= 1 ⭐
  • WORK — REAL array.
    • If ICOMPQ = 0 or 1, the dimension of WORK must be at least
1 + 3*N + 2*N*lg N + 3*N**2
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • If ICOMPQ = 2, the dimension of WORK must be at least
4*N + N**2.
  • IWORK — INTEGER array.
    • If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
6 + 6*N + 5*N*lg N.
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • If ICOMPQ = 2, the dimension of IWORK must be at least
3 + 5*N.
  • INFO — INTEGER.
    • = 0: successful exit 😌
    • < 0: if INFO = -i, the i-th argument had an illegal value 🚨
    • > 0: the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1) 😭

🌈 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 170 of file slaed0.f.

🦋 subroutine zlaed0 🦋

subroutine zlaed0 (integer qsiz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldqs, * ) qstore, integer ldqs, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZLAED0 is used by ZSTEDC. It computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method 🔥

🎀 Purpose 🎀

Using the divide and conquer method, ZLAED0 computes all eigenvalues of a symmetric tridiagonal matrix which is one diagonal block of those from reducing a dense or band Hermitian matrix and corresponding eigenvectors of the dense or band matrix 💖

🌸 Parameters 🌸

  • QSIZ — INTEGER. The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1 👑
  • N — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0 ✨
  • D — DOUBLE PRECISION array, dimension (N). On entry, the diagonal elements of the tridiagonal matrix. On exit, the eigenvalues in ascending order 🎀
  • E — DOUBLE PRECISION array, dimension (N-1). On entry, the off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed 💀
  • Q — COMPLEX*16 array, dimension (LDQ,N). On entry, Q must contain an QSIZ x N matrix whose columns unitarily orthonormal. It is a part of the unitary matrix that reduces the full dense Hermitian matrix to a (reducible) symmetric tridiagonal matrix 🦋
  • LDQ — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) 🌸
  • IWORK — INTEGER array. The dimension of IWORK must be at least
6 + 6*N + 5*N*lg N
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • RWORK — DOUBLE PRECISION array, dimension (1 + 3*N + 2*N*lg N + 3*N**2)
( lg( N ) = smallest integer k
            such that 2^k >= N )
  • QSTORE — COMPLEX*16 array, dimension (LDQS, N). Used to store parts of the eigenvector matrix when the updating matrix multiplies take place 🫶
  • LDQS — INTEGER. The leading dimension of the array QSTORE. LDQS >= max(1,N) ⭐
  • INFO — INTEGER.
    • = 0: successful exit 😌
    • < 0: if INFO = -i, the i-th argument had an illegal value 🚨
    • > 0: the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1) 😭

🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 143 of file zlaed0.f.

🫶 Author 🫶

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