gehd2(3)
💅 gehd2(3) — reduction to Hessenberg, level 2 (yes bestie, THAT gehd2(3), from the Library Functions Manual)
okay so picture this: you've got a big square matrix and it's giving messy. gehd2 is the glow-up routine — it takes your general square matrix and serves upper Hessenberg form, and it does it with the unblocked algorithm (aka the level-2 tea, no blocking, one reflector at a time, very main character energy). Four flavors live here, one for each precision/type combo. No cap. 🌸
✨ SYNOPSIS ✨
🫶 Functions
subroutine cgehd2 (n, ilo, ihi, a, lda, tau, work, info)— CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.subroutine dgehd2 (n, ilo, ihi, a, lda, tau, work, info)— DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.subroutine sgehd2 (n, ilo, ihi, a, lda, tau, work, info)— SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.subroutine zgehd2 (n, ilo, ihi, a, lda, tau, work, info)— ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
💖 literally four icons, same job, different data types. pick your fighter.
🌈 Detailed Description
deadass this page is just the Function Documentation. scroll down bestie, it's all there. 👇
👑 Function Documentation
🦋 subroutine cgehd2 (integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer info)
CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
💜 Purpose:
!>
!> CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
!> by a unitary similarity transformation: Q**H * A * Q = H .
!>the way it just vibes a whole similarity transform into existence. it's giving ✨ linear algebra ✨.
🎀 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 CGEBAL; otherwise they should be
!> set to 1 and N respectively. See Further Details.
!> 1 <= ILO <= IHI <= max(1,N).
!>okay so ILO and IHI are the "already done, don't touch" zone markers — the assumption is that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. Those two usually come from a prior call to CGEBAL; if you're not doing that, just set them to 1 and N. and the constraint is 1 <= ILO <= IHI <= max(1,N), periodt. 🫶
A
!> A is 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.
!>on entry: the n by n general matrix to be reduced. on exit: the upper triangle and first subdiagonal of A get overwritten with the upper Hessenberg matrix H, and everything below the first subdiagonal — together with the array TAU — represents the unitary matrix Q as a product of elementary reflectors. 💅 see Further Details, obviously.
LDA
!> LDA is INTEGER
!> The leading dimension of the array A. LDA >= max(1,N).
!>TAU
!> TAU is COMPLEX array, dimension (N-1)
!> The scalar factors of the elementary reflectors (see Further
!> Details).
!>WORK
!> WORK is COMPLEX array, dimension (N)
!>INFO
!> INFO is INTEGER
!> = 0: successful exit
!> < 0: if INFO = -i, the i-th argument had an illegal value.
!>0 means slay, we're done. anything < 0 means the i-th argument was illegal and INFO told you exactly which one narc'd. 🙈
🌟 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).
!>so Q is a product of (ihi-ilo) elementary reflectors, Q = H(ilo) H(ilo+1) . . . H(ihi-1), and each H(i) has the form H(i) = I - tau * v * v**H. tau is a complex scalar, v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) lands on exit in A(i+2:ihi,i), and tau lands in TAU(i). the little diagram above? that's the glow-up before/after. a = element of the original matrix A, h = modified element of the upper Hessenberg matrix H, and vi = an element of the vector defining H(i). iconic honestly. 📸
Definition at line 148 of file cgehd2.f.
🍓 subroutine dgehd2 (integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
💜 Purpose:
!>
!> DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
!> an orthogonal similarity transformation: Q**T * A * Q = H .
!>same ritual, but real, and it's a transpose (Q**T) instead of a conjugate transpose. know the difference bestie, iykyk. 🦋
🎀 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 DGEBAL; otherwise they should be
!> set to 1 and N respectively. See Further Details.
!> 1 <= ILO <= IHI <= max(1,N).
!>again with the 1:ILO-1 and IHI+1:N already-upper-triangular assumption, and again these come from a previous call — this time to DGEBAL. otherwise 1 and N. and yes, 1 <= ILO <= IHI <= max(1,N). no notes. 💅
A
!> A is 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
!> LDA is INTEGER
!> The leading dimension of the array A. LDA >= max(1,N).
!>TAU
!> TAU is DOUBLE PRECISION array, dimension (N-1)
!> The scalar factors of the elementary reflectors (see Further
!> Details).
!>WORK
!> WORK is DOUBLE PRECISION array, dimension (N)
!>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**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).
!>same deal, except H(i) = I - tau * v * v**T with tau a real scalar and v a real vector. everything else — the (ihi-ilo) reflectors, the zeroed v(1:i), the v(i+1) = 1, the v(ihi+1:n) = 0, the storage of v(i+2:ihi) in A(i+2:ihi,i) and tau in TAU(i) — all identical. the before/after diagram is literally the same and honestly? still obsessed. 🔥
Definition at line 148 of file dgehd2.f.
🧊 subroutine sgehd2 (integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
💜 Purpose:
!>
!> SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
!> an orthogonal similarity transformation: Q**T * A * Q = H .
!>single-precision twin of DGEHD2. same energy, smaller floats. 🌸
🎀 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 SGEBAL; otherwise they should be
!> set to 1 and N respectively. See Further Details.
!> 1 <= ILO <= IHI <= max(1,N).
!>this time the prior call is to SGEBAL — the rest of the vibe is unchanged: 1:ILO-1 and IHI+1:N assumed upper triangular, otherwise just 1 and N, and 1 <= ILO <= IHI <= max(1,N). 🫶
A
!> A is 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
!> 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).
!>WORK
!> WORK is REAL array, dimension (N)
!>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:
...and that's exactly where we're pausing, bestie — Further Details for sgehd2 picks up in part 2. don't go anywhere. 🎀
okay so the matrix Q over here is literally built as a product of (ihi-ilo) elementary reflectors, periodt:
!> Q = H(ilo) H(ilo+1) . . . H(ihi-1).and each H(i) is giving:
!> H(i) = I - tau * v * v**Twhere 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; on exit, v(i+2:ihi) is stored in A(i+2:ihi,i), and tau is stored in TAU(i). no cap 💅
the contents of A are illustrated by this example, with n = 7, ilo = 2, and ihi = 6 — literally the source draws it like this:
!> 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).and fr fr, this one lives at Definition at line 148 of file sgehd2.f. 🫶
🌟 subroutine zgehd2 🌟
subroutine zgehd2 (integer n, integer ilo, integer ihi, complex*16,
dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau,
complex*16, dimension( * ) work, integer info)ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. iconic behavior, honestly 😌
💖 Purpose 💖
ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation:
!> Q**H * A * Q = H .like, that's the whole mission bestie ✨
🌸 Parameters 🌸
N—N is INTEGER. the order of the matrixA.N >= 0.ILO—ILO is INTEGER.IHI—IHI is INTEGER. it is assumed thatAis already upper triangular in rows and columns1:ILO-1andIHI+1:N.ILOandIHIare normally set by a previous call toZGEBAL; otherwise they should be set to1andNrespectively. see Further Details.1 <= ILO <= IHI <= max(1,N).A—A is COMPLEX*16 array, dimension (LDA,N). on entry, thenbyngeneral matrix to be reduced. on exit, the upper triangle and the first subdiagonal ofAare overwritten with the upper Hessenberg matrixH, and the elements below the first subdiagonal, with the arrayTAU, represent the unitary matrixQas a product of elementary reflectors. see Further Details.LDA—LDA is INTEGER. the leading dimension of the arrayA.LDA >= max(1,N).TAU—TAU is COMPLEX*16 array, dimension (N-1). the scalar factors of the elementary reflectors (see Further Details).WORK—WORK is COMPLEX*16 array, dimension (N).INFO—INFO is INTEGER.= 0: successful exit.< 0: ifINFO = -i, thei-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**Hwhere 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; on exit, v(i+2:ihi) is stored in A(i+2:ihi,i), and tau is stored in TAU(i). slay ✨
the contents of A are illustrated by this 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).and this one is also at Definition at line 148 of file zgehd2.f. 🦋
Author (of the page itself): Generated automatically by Doxygen for LAPACK from the source code.
okay that's the whole gehd2(3) moment bestie — LAPACK Version 3.12.0, footer source/date tag gehd2(3), love u bye 💖✨🎀
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