SRC/ssytrd.f(3)

Okay bestie, this is SRC/ssytrd.f(3) from the Library Functions Manual — and yes, ssytrd is the main character here. 🫶

🎀 NAME 🎀

SRC/ssytrd.f — that's the file, no cap. It's the manual page for the ssytrd routine. ✨

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

• subroutine ssytrd (uplo, n, a, lda, d, e, tau, work, lwork, info) — SSYTRD

💖 Function/Subroutine Documentation 💖

subroutine ssytrd (character uplo, integer n, real, dimension( lda, * ) a,
    integer lda, real, dimension( * ) d, real, dimension( * ) e, real,
    dimension( * ) tau, real, dimension( * ) work, integer lwork, integer
    info)

SSYTRD 💅

🦋 Purpose 🦋

Okay so SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation:

Q**T * A * Q = T.

🌸 Parameters 🌸

• UPLO — UPLO is CHARACTER*1
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER; the order of the matrix A. N >= 0.
• A — A is REAL array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, 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).
• D — D is REAL array, dimension (N); the diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
• E — E is REAL array, dimension (N-1); the off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
• TAU — TAU is REAL array, dimension (N-1); the scalar factors of the elementary reflectors (see Further Details).
• 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 dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= 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
  • = 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 🔥

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(n-1) . . . H(2) H(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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

📍 Definition at line 191 of file ssytrd.f.

🖥️ Author 🖥️

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Sign-off time, bestie: this footer situation is LAPACK Version 3.12.0 with source tag SRC/ssytrd.f(3). The original footer doesn't print a date line, so that's all we got — periodt. 💜🌈