gghd3(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🌸 NAME 🌸

okay so this is gghd3(3) from the Library Functions Manual, and the NAME line is literally gghd3 - gghd3: reduction to Hessenberg, level 3. bestie, it's giving “we reduce matrices and we do it at level 3,” periodt 💅

✨ SYNOPSIS ✨

Functions, aka the whole squad — same energy, different precision:

subroutine cgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)
    CGGHD3
subroutine dgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)
    DGGHD3
subroutine sgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)
    SGGHD3
subroutine zgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
    ldz, work, lwork, info)
    ZGGHD3

💖 Detailed Description 💖

This is the part where the page points you at the actual function docs. The main character energy lives under Function Documentation, and yes, the details are lowkey the whole point 🫶

🧚 Function Documentation 🧚

✨ subroutine cgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info) — CGGHD3 ✨

Purpose

okay so CGGHD3 takes a pair of complex matrices (A,B) and reduces them to generalized upper Hessenberg form using unitary transformations. A is a general matrix, and B is upper triangular. The generalized eigenvalue problem is:

A*x = lambda*B*x,

and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:

Q**H*A*Z = H

and transforms B to another upper triangular matrix T:

Q**H*B*Z = T

in order to reduce the problem to its standard form:

H*y = lambda*T*y

where y = Z**H*x.

The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that:

Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H

Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H

If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then CGGHD3 reduces the original problem to generalized Hessenberg form.

This is a blocked variant of CGGHRD, using matrix-matrix multiplications for parts of the computation to enhance performance. iconic behavior, no cap 💅

🌸 Parameters 🌸

  • COMPQ — CHARACTER*1.
    • = 'N': do not compute Q;
    • = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned;
    • = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.
  • COMPZ — CHARACTER*1.
    • = 'N': do not compute Z;
    • = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned;
    • = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • ILO — INTEGER.
  • IHI — INTEGER.

ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  • A — COMPLEX array, dimension (LDA, N).

On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — COMPLEX array, dimension (LDB, N).

On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero.

  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • Q — COMPLEX array, dimension (LDQ, N).

On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

  • LDQ — INTEGER. The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
  • Z — COMPLEX array, dimension (LDZ, N).

On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
  • WORK — COMPLEX array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  • LWORK — INTEGER. The length of the array WORK. LWORK >= 1. For optimum performance LWORK >= 6*N*NB, where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • 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.

🦋 Further Details 🦋

This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008).

Definition at line 229 of file cgghd3.f.

✨ subroutine dgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info) — DGGHD3 ✨

Purpose

DGGHD3 does the same glow-up for a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations. A is a general matrix, and B is upper triangular. The generalized eigenvalue problem is:

A*x = lambda*B*x,

and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:

Q**T*A*Z = H

and transforms B to another upper triangular matrix T:

Q**T*B*Z = T

in order to reduce the problem to its standard form:

H*y = lambda*T*y

where y = Z**T*x.

The orthogonal matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that:

Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then DGGHD3 reduces the original problem to generalized Hessenberg form.

This is a blocked variant of DGGHRD, using matrix-matrix multiplications for parts of the computation to enhance performance. fr fr, it ate 💅

🌸 Parameters 🌸

  • COMPQ — CHARACTER*1.
    • = 'N': do not compute Q;
    • = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned;
    • = 'V': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.
  • COMPZ — CHARACTER*1.
    • = 'N': do not compute Z;
    • = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned;
    • = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • ILO — INTEGER.
  • IHI — INTEGER.

ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  • A
  • A — a DOUBLE PRECISION array, dimension (LDA, N). On entry it's the N-by-N general matrix you're reducing, aka the whole reason we're here 💅. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero. so like, your original A? gone. it ate and left no crumbs 😭
  • LDA — INTEGER, the leading dimension of the array A. Needs LDA >= max(1,N). disrespect this number and the vibes will be rancid fr.
  • B — DOUBLE PRECISION array, dimension (LDB, N). On entry: the N-by-N upper triangular matrix B. On exit: the upper triangular matrix T = Q**T B Z, and the elements below the diagonal are set to zero ✨. she stays triangular, that's her whole brand.
  • LDB — INTEGER, the leading dimension of the array B. LDB >= max(1,N). that's the rule, bestie 🫶
  • Q — DOUBLE PRECISION array, dimension (LDQ, N). On entry, if COMPQ = 'V', this holds the orthogonal matrix Q1 — typically from the QR factorization of B, iykyk. On exit, if COMPQ='I' it holds the orthogonal matrix Q, and if COMPQ = 'V' it holds the product Q1*Q. Not referenced if COMPQ='N' — she's off duty 💤
  • LDQ — INTEGER, the leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. different rules for different girls 🌸
  • Z — DOUBLE PRECISION array, dimension (LDZ, N). On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I' the orthogonal matrix Z, and if COMPZ = 'V' the product Z1*Z. Not referenced if COMPZ='N'. iconic, no notes 👑
  • LDZ — INTEGER, the leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. same energy as LDQ, it's giving sisterhood 🎀
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's the scratch queen, treat her right 🧚
  • LWORK — INTEGER, the length of the array WORK. LWORK >= 1. For optimum performance LWORK >= 6*N*NB, where NB is the optimal blocksize — like, technically 1 works, but that's not the slay we're going for 🌈

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. so -1 is the "just tell me what you want from me" energy 💅

  • INFO — INTEGER. Exit report:
    • = 0: successful exit. we love to see it 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. the callout is COMING 💀

🦋 Author 🦋

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

💖 Further Details 💖

This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008). historic behaviour, honestly ✨

Definition at line 228 of file dgghd3.f.

🦋 subroutine sgghd3 🦋

Same whole vibe, just in REAL. no cap 🫶

subroutine sgghd3 (character compq, character compz, integer n, integer ilo,
    integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension(
    ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real,
    dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer
    lwork, integer info)

SGGHD3

🌸 Purpose 🌸

SGGHD3 reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is

A*x = lambda*B*x,

and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation. classic move 🎀

This subroutine simultaneously reduces A to a Hessenberg matrix H:

Q**T*A*Z = H

and transforms B to another upper triangular matrix T:

Q**T*B*Z = T

in order to reduce the problem to its standard form

H*y = lambda*T*y

where y = Z**T*x.

The orthogonal matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that

     Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

     Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then SGGHD3 reduces the original problem to generalized Hessenberg form. that's the payoff bestie 💖

This is a blocked variant of SGGHRD, using matrix-matrix multiplications for parts of the computation to enhance performance. blocked = faster = she's built different 🔥

✨ Parameters ✨

  • COMPQ — CHARACTER*1.
    • = 'N': do not compute Q; (she's not coming to the function 😔)
    • = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned;
    • = 'V': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.
  • COMPZ — CHARACTER*1.
    • = 'N': do not compute Z;
    • = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned;
    • = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.
  • N — INTEGER. The order of the matrices A and B. N >= 0. zero is allowed, she can be nothing, that's valid 🦋
  • ILO — INTEGER. (the vibe check partner of IHI, see below 💅)
  • IHI — INTEGER.

ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to SGGBAL; otherwise they should be set to 1 and N respectively. Also: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. yes the empty case is flipped, no we do not make the rules 😭

  • A — REAL array, dimension (LDA, N). On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero. poof. gone. iconic behaviour 💀
  • LDA — INTEGER, the leading dimension of the array A. LDA >= max(1,N). mandatory ⭐
  • B — REAL array, dimension (LDB, N). On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero. she is cleansed ✨
  • LDB — INTEGER, the leading dimension of the array B. LDB >= max(1,N). ty bestie 🫶
  • Q — REAL array, dimension (LDQ, N). On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'. she said "not today" 💅
  • LDQ — INTEGER, the leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. conditional queen 🎀
  • Z — REAL array, dimension (LDZ, N). On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'. main character of the right-hand side 🌟
  • LDZ — INTEGER, the leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. same rules, different letter 🌈
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. the workspace, our unsung hero 🧚
  • LWORK — INTEGER, the length of the array WORK. LWORK >= 1. For optimum performance LWORK >= 6*N*NB, where NB is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. so you can just ask "how much do you need from me" and she'll tell you, no nagging 😌

  • INFO — INTEGER.
    • = 0: successful exit. 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. somebody's getting read for filth 💀

🦋 Author 🦋

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

💜 Further Details 💜

This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008). citation slay 🎀

Definition at line 228 of file sgghd3.f.

💅 subroutine zgghd3 💅

now in COMPLEX*16. the plot thickens 👀

subroutine zgghd3 (character compq, character compz, integer n, integer ilo,
    integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16,
    dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q,
    integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, complex*16,
    dimension( * ) work, integer lwork, integer info)

ZGGHD3

👑 Purpose 👑

🌸 Parameters 🌸

Okay bestie, this is where we get into the actual arguments. Every single one matters, so pay attention 💅

COMPQ

COMPQ is CHARACTER*1. It tells the routine what to do with the unitary matrix Q:

  • = 'N': do not compute Q;
  • = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned;
  • = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.

COMPZ

COMPZ is CHARACTER*1. Same vibe as COMPQ, but for Z:

  • = 'N': do not compute Z;
  • = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned;
  • = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.

N

N is INTEGER. It's the order of the matrices A and B. Constraint: N >= 0.

ILO

ILO is INTEGER.

IHI

IHI is INTEGER.

ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGGBAL; otherwise they should be set to 1 and N respectively. Constraints: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

A is COMPLEX*16 array, dimension (LDA, N). On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

LDA

LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,N).

B

B is COMPLEX*16 array, dimension (LDB, N). On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero.

LDB

LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N).

Q

Q is COMPLEX*16 array, dimension (LDQ, N). On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

LDQ

LDQ is INTEGER. The leading dimension of the array Q. Constraint: LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

Z is COMPLEX*16 array, dimension (LDZ, N). On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

LDZ

LDZ is INTEGER. The leading dimension of the array Z. Constraint: LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER. The length of the array WORK. Constraint: LWORK >= 1. For optimum performance LWORK >= 6*N*NB, where NB is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER. Exit status:

  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

🎀 Author 🎀

This iconic routine was brought to you by:

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

💜 Further Details: 💜

This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008).

Definition at line 225 of file zgghd3.f.

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole page, bestie! This lil manual page is from LAPACK Version 3.12.0 and it's the gghd3(3) page. Slay 💅✨

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