SRC/dsbtrd.f(3)

💅 SRC/dsbtrd.f — like, the actual manual page bestie

okay so real talk: this is SRC/dsbtrd.f, straight out of the Library Functions Manual, and it's lowkey one of those files that sounds scary but is honestly giving main character energy once you get it. Let's go 🫶

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✨ NAME ✨

SRC/dsbtrd.f — that's it, that's the name, no cap. She's a Fortran source file and she knows it 💅

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🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine dsbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) — this is her, this is the girl. Also goes by DSBTRD ✨

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🦋 Function/Subroutine Documentation 🦋

subroutine dsbtrd (character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer info)

aka DSBTRD, the full signature, verbatim, don't @ me 😌

🎀 Purpose 🎀

!>
!> DSBTRD reduces a real symmetric band matrix A to symmetric
!> tridiagonal form T by an orthogonal similarity transformation:
!> Q**T * A * Q = T.
!>

so basically: symmetric band matrix in, symmetric tridiagonal matrix out, and the whole thing happens via an orthogonal similarity transformation. Q**T * A * Q = T and she ate 🔥

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💜 Parameters 💜

VECT

!>          VECT is CHARACTER*1
!>          = 'N':  do not form Q;
!>          = 'V':  form Q;
!>          = 'U':  update a matrix X, by forming X*Q.
!>

VECT is CHARACTER1* and it's literally picking your vibe: 'N' = do not form Q, 'V' = form Q, 'U' = update a matrix X, by forming X*Q. pick your fighter 🎀

UPLO

!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>

UPLO is CHARACTER1*: 'U' means the Upper triangle of A is stored, 'L' means the Lower triangle of A is stored. up or down, choose wisely ✨

N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

N is INTEGER, the order of the matrix A. Constraint: N >= 0. periodt 💅

KD

!>          KD is INTEGER
!>          The number of superdiagonals of the matrix A if UPLO = 'U',
!>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
!>

KD is INTEGER — the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KD >= 0. she's flexible like that 🦋

AB

!>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
!>          On entry, the upper or lower triangle of the symmetric band
!>          matrix A, stored in the first KD+1 rows of the array.  The
!>          j-th column of A is stored in the j-th column of the array AB
!>          as follows:
!>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
!>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
!>          On exit, the diagonal elements of AB are overwritten by the
!>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
!>          elements on the first superdiagonal (if UPLO = 'U') or the
!>          first subdiagonal (if UPLO = 'L') are overwritten by the
!>          off-diagonal elements of T; the rest of AB is overwritten by
!>          values generated during the reduction.
!>

AB is a DOUBLE PRECISION array, dimension (LDAB,N) and honestly she's the main character here 👑

On entry she's holding the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A lives in the j-th column of the array AB like this:

• if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
• if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit? The diagonal elements of AB get overwritten by the diagonal elements of the tridiagonal matrix T. And if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') get overwritten by the off-diagonal elements of T. The rest of AB is overwritten by values generated during the reduction. she's been through a whole glow-up, tbh 💀

LDAB

!>          LDAB is INTEGER
!>          The leading dimension of the array AB.  LDAB >= KD+1.
!>

LDAB is INTEGER, the leading dimension of the array AB. Constraint: LDAB >= KD+1. do not disrespect her ✋

D

!>          D is DOUBLE PRECISION array, dimension (N)
!>          The diagonal elements of the tridiagonal matrix T.
!>

D is a DOUBLE PRECISION array, dimension (N) — the diagonal elements of the tridiagonal matrix T. she's the main diagonal and she's thriving 🎀

E

!>          E is DOUBLE PRECISION array, dimension (N-1)
!>          The off-diagonal elements of the tridiagonal matrix T:
!>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
!>

E is a DOUBLE PRECISION array, dimension (N-1) — the off-diagonal elements of the tridiagonal matrix T:

• E(i) = T(i,i+1) if UPLO = 'U'
• E(i) = T(i+1,i) if UPLO = 'L'

she's off the diagonal and she's iconic ✨

Q

!>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
!>          On entry, if VECT = 'U', then Q must contain an N-by-N
!>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
!>
!>          On exit:
!>          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
!>          if VECT = 'U', Q contains the product X*Q;
!>          if VECT = 'N', the array Q is not referenced.
!>

Q is a DOUBLE PRECISION array, dimension (LDQ,N) 💜

On entry: if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. (no pressure bestie)

On exit, the lore:

• if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
• if VECT = 'U', Q contains the product X*Q;
• if VECT = 'N', the array Q is not referenced. she's literally not even invited to the party 🙈

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.
!>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
!>

LDQ is INTEGER, the leading dimension of the array Q. Constraints: LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. read the room 📏

WORK

!>          WORK is DOUBLE PRECISION array, dimension (N)
!>

WORK is a DOUBLE PRECISION array, dimension (N). that's it. she works. 💅

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>

INFO is INTEGER — the vibe check at the end:

• = 0: successful exit 🎉
• < 0: if INFO = -i, the i-th argument had an illegal value 😭

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👑 Author 👑

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

the whole squad, contributing, we love to see it 🫶

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🌈 Further Details: 🌈

!>
!>  Modified by Linda Kaufman, Bell Labs.
!>

shoutout Linda Kaufman at Bell Labs, she modified it, we stan 💖

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⭐ Definition at line 161 of file dsbtrd.f. ⭐

literally line 161, in dsbtrd.f, that's where she lives 🏡

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code. no human hands, just vibes and a build script 🤖✨

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ok that's the whole page, nothing skipped, nothing dropped, every flag and constraint and line number intact. she's thorough 💅

LAPACK · Version 3.12.0 · SRC/dsbtrd.f(3) — mwah, that's the footer, and that's the tea 🍓 byeee bestie, go reduce some band matrices 🦋💖