TESTING/MATGEN/zlarot.f(3)

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💅 TESTING/MATGEN/zlarot.f(3) — Library Functions Manual

okay bestie, welcome to the zlarot page. This is the Library Functions Manual entry for TESTING/MATGEN/zlarot.f(3), and we are gonna get into the Givens-rotation of it all without losing a single deet. 🎀

✨ NAME ✨

TESTING/MATGEN/zlarot.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

  • subroutine zlarot (lrows, lleft, lright, nl, c, s, a, lda, xleft, xright) — ZLAROT

💖 Function/Subroutine Documentation 💖

subroutine zlarot (logical lrows, logical lleft, logical lright, integer nl,
    complex*16 c, complex*16 s, complex*16, dimension( * ) a, integer lda,
    complex*16 xleft, complex*16 xright)

ZLAROT

🔥 Purpose

ZLAROT applies a (Givens) rotation to two adjacent rows or columns, where one element of the first and/or last column/row is handled 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. periodt.

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 ZLAROT 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.

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

General dense matrix:

CALL ZLAROT(.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 ZLAROT( .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 ZLAROT( .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 ZLAROT( .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 ZLAROT( .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 )

💅 Arguments

  • LROWS — LOGICAL. If .TRUE., then ZLAROT 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 — COMPLEX*16. 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 (not conjugated) thereof is applied from the right. Note that in contrast to the output of ZROTG or to most versions of ZROT, both C and S are complex. For a Givens rotation, |C|**2 + |S|**2 should be 1, but this is not checked. Not modified.

  • A — COMPLEX*16 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, HE, 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, HB, 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 ZLAROT, 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 — COMPLEX*16. 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 — COMPLEX*16. 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 227 of file zlarot.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

💖 That’s the whole zlarot moment, bestie. Source/version vibes: LAPACK, Version 3.12.0, from the TESTING/MATGEN/zlarot.f(3) page. The footer doesn’t show a date, so I’m not inventing one — periodt. 🎀✨

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