SGGHRD(3)

🦋 SGGHRD(3) 🦋

okay bestie, welcome to the SGGHRD(3) page from the Library Functions Manual. Highkey the file it’s all about is /home/abui...RC/sgghrd.f — yes, that truncated-looking path is literally what’s printed, no cap 💅.

🌸 NAME 🌸

/home/abui...RC/sgghrd.f — that’s the name/header tea, and it’s giving main character energy ✨.

✨ SYNOPSIS ✨

💫 Functions/Subroutines 💫

This is the subroutine bestie, fr fr:

       subroutine SGGHRD (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z,
           ldz, info)
           SGGHRD

🎀 Function/Subroutine Documentation 🎀

Here’s the full typed signature so you know exactly what’s on the menu, no guessing:

   subroutine SGGHRD (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, integer info)
       SGGHRD

🔥 Purpose 🔥

okay so SGGHRD 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. iconic behaviour, honestly 👑.

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 SGGHRD reduces the original problem to generalized Hessenberg form. periodt 💅.

🌈 Parameters 🌈

vibe check on the parameters, bestie — every single one is here, ngl:

• 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 SGGBAL; 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 — 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.
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• 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.
• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• 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'.
• LDQ — INTEGER. The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
• 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'.
• LDZ — INTEGER. The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
• INFO — INTEGER
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author 👑

the author credits are giving collaborative queen energy:

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

🫶 Further Details 🫶

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and Van Loan (Johns Hopkins Press.)

Definition at line 205 of file sgghrd.f.

👑 Author 👑 (the Doxygen one)

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie that’s the whole SGGHRD(3) page 💖 Signing off with the footer tea: LAPACK Version 3.12.0, /home/abui...RC/sgghrd.f(3) — and yeah, no date is printed on this one, so we’re not inventing one 🌸🫶. iykyk.