SRC/dsytrd.f(3)

🌸 NAME 🌸

okay so this is SRC/dsytrd.f, bestie — straight out of the Library Functions Manual (section 3, as the header vibe implies). It's the page for the dsytrd subroutine, and we're gonna get into it. periodt. 💅

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

• subroutine dsytrd (uplo, n, a, lda, d, e, tau, work, lwork, info) — the whole squad of args, right here. DSYTRD is the icon we're documenting. no cap.

💖 Function/Subroutine Documentation 💖

subroutine dsytrd (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD — yes, she's the main character.

🎀 Purpose 🎀

okay so DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. literally that's the whole assignment, and she lowkey ate. 🔥

🌈 Parameters 🌈

• UPLO — CHARACTER*1. This tells us which triangle of A is stored:
  • 'U': upper triangle of A is stored;
  • 'L': lower triangle of A is stored.
• N — INTEGER. The order of the matrix A. N >= 0. no negative orders, bestie — vibe check passed.
• A — DOUBLE PRECISION array, dimension (LDA,N). Here's the tea:
  • On entry, this is 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 over-written 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. 👀 iykyk.
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). respect the leading dimension, periodt. ngl, this one matters.
• D — DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). cute.
• E — DOUBLE PRECISION 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'. iconic.
• TAU — DOUBLE PRECISION array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details). she's giving reflector lore. 💅
• WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. obsessed with this little messenger. 🫶
• LWORK — 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 — INTEGER. The exit report:
  • = 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 🦋

okay so here's the reflector lore, bestie. ngl, it's real. 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 dsytrd.f.

🤖 Author (again, but make it Doxygen) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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