TESTING/MATGEN/dlarot.f(3)

🎀 TESTING/MATGEN/dlarot.f(3) 🎀

okay so this page is from the Library Functions Manual, and it's all about DLAROT — the LAPACK subroutine that applies Givens rotations to adjacent rows/columns in matrix formats where some entries live outside the array. it’s giving “I can rotate your matrix, but only the parts I can actually touch” 💅✨

✨ NAME ✨

TESTING/MATGEN/dlarot.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine dlarot (lrows, lleft, lright, nl, c, s, a, lda, xleft, xright) — DLAROT. that’s the summoning circle, bestie 🫶

🦋 Function/Subroutine Documentation 🦋

subroutine dlarot (logical lrows, logical lleft, logical lright, integer nl,
     double precision c, double precision s, double precision, dimension( * )
     a, integer lda, double precision xleft, double precision xright)

DLAROT — iconic name, no notes 👑

💜 Purpose 💜

DLAROT applies a (Givens) rotation to two adjacent rows or columns, where one element of the first and/or last column/row for use on matrices stored in some format other than GE, so that elements of the matrix may be used or modified for which no array element is provided. yes, it’s that specific 😭

One example is a symmetric matrix in SB format (bandwidth=4), for which UPLO='L': two adjacent rows will have the format:

    row j:     C> C> C> C> C> .  .  .  .
    row j+1:      C> C> C> C> C> .  .  .  .

'*' indicates elements for which storage is provided, '.' indicates elements for which no storage is provided, but are not necessarily zero; their values are determined by symmetry. ' ' indicates elements which are necessarily zero, and have no storage provided.

Those columns which have two '*'s can be handled by DROT. Those columns which have no '*'s can be ignored, since as long as the Givens rotations are carefully applied to preserve symmetry, their values are determined. Those columns which have one '*' have to be handled separately, by using separate variables and:

    row j:     C> C> C> C> C> p  .  .  .
    row j+1:   q  C> C> C> C> C> .  .  .  .

The element p would have to be set correctly, then that column is rotated, setting p to its new value. The next call to DLAROT would rotate columns j and j+1, using p, and restore symmetry. The element q would start out being zero, and be made non-zero by the rotation. Later, rotations would presumably be chosen to zero q out. slay, symmetry preserved 💖

Typical Calling Sequences: rotating the i-th and (i+1)-st rows. ------- ------- ---------

General dense matrix:

              CALL DLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
                      A(i,1),LDA, DUMMY, DUMMY)

General banded matrix in GB format:

              j = MAX(1, i-KL )
              NL = MIN( N, i+KU+1 ) + 1-j
              CALL DLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
                      A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )

[ note that i+1-j is just MIN(i,KL+1) ]

Symmetric banded matrix in SY format, bandwidth K, lower triangle only:

              j = MAX(1, i-K )
              NL = MIN( K+1, i ) + 1
              CALL DLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
                      A(i,j), LDA, XLEFT, XRIGHT )

Same, but upper triangle only:

              NL = MIN( K+1, N-i ) + 1
              CALL DLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
                      A(i,i), LDA, XLEFT, XRIGHT )

Symmetric banded matrix in SB format, bandwidth K, lower triangle only:

              [ same as for SY, except:]
                  . . . .
                      A(i+1-j,j), LDA-1, XLEFT, XRIGHT )

[ note that i+1-j is just MIN(i,K+1) ]

Same, but upper triangle only:

                   . . .
                      A(K+1,i), LDA-1, XLEFT, XRIGHT )

Rotating columns is just the transpose of rotating rows, except for GB and SB: (rotating columns i and i+1)

GB:

              j = MAX(1, i-KU )
              NL = MIN( N, i+KL+1 ) + 1-j
              CALL DLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
                      A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )

[note that KU+j+1-i is just MAX(1,KU+2-i)]

SB: (upper triangle)

                   . . . . . .
                      A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )

SB: (lower triangle)

                   . . . . . .
                      A(1,i),LDA-1, XTOP, XBOTTM )

🍓 Parameters 🍓

• LROWS — LOGICAL. If .TRUE., then DLAROT will rotate two rows. If .FALSE., then it will rotate two columns. Not modified.
• LLEFT — LOGICAL. If .TRUE., then XLEFT will be used instead of the corresponding element of A for the first element in the second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.). If .FALSE., then the corresponding element of A will be used. Not modified.
• LRIGHT — LOGICAL. If .TRUE., then XRIGHT will be used instead of the corresponding element of A for the last element in the first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.). If .FALSE., then the corresponding element of A will be used. Not modified.
• NL — INTEGER. The length of the rows (if LROWS=.TRUE.) or columns (if LROWS=.FALSE.) to be rotated. If XLEFT and/or XRIGHT are used, the columns/rows they are in should be included in NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at least 2. The number of rows/columns to be rotated exclusive of those involving XLEFT and/or XRIGHT may not be negative, i.e., NL minus how many of LLEFT and LRIGHT are .TRUE. must be at least zero; if not, XERBLA will be called. Not modified.
• C, S — DOUBLE PRECISION. Specify the Givens rotation to be applied. If LROWS is true, then the matrix

( c  s )
(-s  c )

is applied from the left; if false, then the transpose thereof is applied from the right. For a Givens rotation, C**2 + S**2 should be 1, but this is not checked. Not modified.

bestie, “should be 1, but this is not checked” is doing a lot of heavy lifting 😭

• A — DOUBLE PRECISION array. The array containing the rows/columns to be rotated. The first element of A should be the upper left element to be rotated. Read and modified.
• LDA — INTEGER. The leading dimension of A. If A contains a matrix stored in GE or SY format, then this is just the leading dimension of A as dimensioned in the calling routine. If A contains a matrix stored in band (GB or SB) format, then this should be one less than the leading dimension used in the calling routine. Thus, if A were dimensioned A(LDA,*) in DLAROT, then A(1,j) would be the j-th element in the first of the two rows to be rotated, and A(2,j) would be the j-th in the second, regardless of how the array may be stored in the calling routine. [A cannot, however, actually be dimensioned thus, since for band format, the row number may exceed LDA, which is not legal FORTRAN.] If LROWS=.TRUE., then LDA must be at least 1, otherwise it must be at least NL minus the number of .TRUE. values in XLEFT and XRIGHT. Not modified.
• XLEFT — DOUBLE PRECISION. If LLEFT is .TRUE., then XLEFT will be used and modified instead of A(2,1) (if LROWS=.TRUE.) or A(1,2) (if LROWS=.FALSE.). Read and modified.
• XRIGHT — DOUBLE PRECISION. If LRIGHT is .TRUE., then XRIGHT will be used and modified instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1) (if LROWS=.FALSE.). Read and modified.

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 224 of file dlarot.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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mwah 💋 and that’s the whole page, bestie. the footer says LAPACK, Version 3.12.0, and TESTING/MATGEN/dlarot.f(3) — no cap, we got every last detail 🎀💖✨