gehrd(3)

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version 3.12.1 · 9 versions
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💖 gehrd(3) — the "reduction to Hessenberg" glow-up, bestie ✨

okay so this is gehrd(3), living its best life in the Library Functions Manual, and it's genuinely one of those pages that's giving "four sisters, same personality" energy. Four subroutines, same signature, different precision flavours. Let's get into it 💅

🌸 NAME 🌸

gehrd — reduction to Hessenberg. That's the whole name, that's the whole vibe, periodt. 👑

✨ SYNOPSIS ✨

🧚 Functions 🧚

  • subroutine cgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info) — the CGEHRD one 💜
  • subroutine dgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info) — the DGEHRD one 🦋
  • subroutine sgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info) — the SGEHRD one 🎀
  • subroutine zgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info) — the ZGEHRD one 🌈

notice how every single one has the exact same argument list? yeah. consistency is mother. 🫶

💫 Detailed Description 💫

the overview section here is lowkey doing the absolute bare minimum — it's just vibes and a heading. the actual tea is all under Function Documentation below, where each subroutine gets its full main-character moment. keep scrolling, bestie. ⭐

👑 Function Documentation 👑

💅 cgehrd 💅

subroutine cgehrd (integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CGEHRD ✨

🌟 Purpose 🌟

so CGEHRD takes a complex general matrix A and reduces it to upper Hessenberg form H by an unitary similarity transformation. (yes, "an unitary" — that's literally what it says, we don't fix it, we respect it 💅) and the transformation is this exact thing right here:

Q**H * A * Q = H .

ate. 🔥

🎀 Parameters 🎀

  • N — INTEGER. The order of the matrix A. N >= 0. 💖
  • ILO — INTEGER. that's genuinely the entire description, no cap 😭 the lore lives in IHI, keep reading.
  • IHI — INTEGER. 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 CGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. And the rules are: 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 elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. 👀
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🎀
  • TAU — COMPLEX array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. ✨
  • 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 >= max(1,N). For good performance, LWORK should generally be larger. 🫶 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.

(icons, all of them 👑)

🔥 Further Details 🔥

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). 🌸

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). 🦋

This file is a slight modification of LAPACK-3.0's CGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See CLAHR2.) 🧚

Definition at line 166 of file cgehrd.f. 📍

💜 dgehrd 💜

subroutine dgehrd (integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DGEHRD ✨

🌟 Purpose 🌟

DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation. (orthogonal this time, not unitary — glow-up, but make it real 💅)

Q**T * A * Q = H .

🎀 Parameters 🎀

  • N — INTEGER. The order of the matrix A. N >= 0. 💖
  • ILO — INTEGER. again, that's the whole description. we move. 😭
  • IHI — INTEGER. 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 DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. Also: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 🦋
  • A — DOUBLE PRECISION 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 elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. 👀
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🎀
  • TAU — DOUBLE PRECISION array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. ✨
  • WORK — DOUBLE PRECISION 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 >= max(1,N). For good performance, LWORK should generally be larger. 🫶 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 🔥

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). 🌸

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). 🦋

This file is a slight modification of LAPACK-3.0's DGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See DLAHR2.) 🧚

Definition at line 166 of file dgehrd.f. 📍

🎀 sgehrd 🎀

subroutine sgehrd (integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SGEHRD ✨

🌟 Purpose 🌟

SGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation. same energy as DGEHRD, different precision paperwork 💅

Q**T * A * Q = H .

🎀 Parameters 🎀

  • N — INTEGER. The order of the matrix A. N >= 0. 💖
  • ILO — INTEGER. (yes, the description really is just that. iconic minimalism.) 🙈
  • IHI — INTEGER. 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 SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. And: 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 🦋
  • A — A is REAL array, dimension (LDA,N). On entry, it's 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 elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. 🫶
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
  • TAU — TAU is REAL array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. ✨
  • WORK — WORK is REAL 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. LWORK >= max(1,N). For good performance, LWORK should generally be larger.
🚨 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. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 😭

Author 🎀

These legends wrote it:

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

Further Details: 📐

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). 🧚

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). 💜

This file is a slight modification of LAPACK-3.0's SGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See SLAHR2.)

📌 Definition at line 166 of file sgehrd.f.

💜 subroutine zgehrd — the COMPLEX*16 sibling

   subroutine zgehrd (integer n, integer ilo, integer ihi, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau,
       complex*16, dimension( * ) work, integer lwork, integer info)

ZGEHRD 💅

Purpose 🎯

ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q**H * A * Q = H.

Parameters 🧮

  • N — N is INTEGER. The order of the matrix A. N >= 0. ✨
  • ILO — ILO is INTEGER. 🫶
  • IHI — IHI is INTEGER. 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 ZGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 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 elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. 💖
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
  • TAU — TAU is COMPLEX*16 array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. ✨
  • 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. LWORK >= max(1,N). For good performance, LWORK should generally be larger.
🚨 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. = 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: 📐

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). 🧚

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). 💜

This file is a slight modification of LAPACK-3.0's ZGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See ZLAHR2.)

📌 Definition at line 166 of file zgehrd.f.

Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

🎀 That's the whole gehrd(3) glow-up, bestie — source/version footer stamp: LAPACK, Version 3.12.0, tag gehrd(3). Go reduce something and slay your Hessenberg. 💅✨
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