LAPACK 3.3.1 Linear Algebra PACKage

# sqrt02.f

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```00001       SUBROUTINE SQRT02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
00002      \$                   RWORK, RESULT )
00003 *
00004 *  -- LAPACK test routine (version 3.1) --
00005 *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       INTEGER            K, LDA, LWORK, M, N
00010 *     ..
00011 *     .. Array Arguments ..
00012       REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
00013      \$                   R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
00014      \$                   WORK( LWORK )
00015 *     ..
00016 *
00017 *  Purpose
00018 *  =======
00019 *
00020 *  SQRT02 tests SORGQR, which generates an m-by-n matrix Q with
00021 *  orthonornmal columns that is defined as the product of k elementary
00022 *  reflectors.
00023 *
00024 *  Given the QR factorization of an m-by-n matrix A, SQRT02 generates
00025 *  the orthogonal matrix Q defined by the factorization of the first k
00026 *  columns of A; it compares R(1:n,1:k) with Q(1:m,1:n)'*A(1:m,1:k),
00027 *  and checks that the columns of Q are orthonormal.
00028 *
00029 *  Arguments
00030 *  =========
00031 *
00032 *  M       (input) INTEGER
00033 *          The number of rows of the matrix Q to be generated.  M >= 0.
00034 *
00035 *  N       (input) INTEGER
00036 *          The number of columns of the matrix Q to be generated.
00037 *          M >= N >= 0.
00038 *
00039 *  K       (input) INTEGER
00040 *          The number of elementary reflectors whose product defines the
00041 *          matrix Q. N >= K >= 0.
00042 *
00043 *  A       (input) REAL array, dimension (LDA,N)
00044 *          The m-by-n matrix A which was factorized by SQRT01.
00045 *
00046 *  AF      (input) REAL array, dimension (LDA,N)
00047 *          Details of the QR factorization of A, as returned by SGEQRF.
00048 *          See SGEQRF for further details.
00049 *
00050 *  Q       (workspace) REAL array, dimension (LDA,N)
00051 *
00052 *  R       (workspace) REAL array, dimension (LDA,N)
00053 *
00054 *  LDA     (input) INTEGER
00055 *          The leading dimension of the arrays A, AF, Q and R. LDA >= M.
00056 *
00057 *  TAU     (input) REAL array, dimension (N)
00058 *          The scalar factors of the elementary reflectors corresponding
00059 *          to the QR factorization in AF.
00060 *
00061 *  WORK    (workspace) REAL array, dimension (LWORK)
00062 *
00063 *  LWORK   (input) INTEGER
00064 *          The dimension of the array WORK.
00065 *
00066 *  RWORK   (workspace) REAL array, dimension (M)
00067 *
00068 *  RESULT  (output) REAL array, dimension (2)
00069 *          The test ratios:
00070 *          RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS )
00071 *          RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )
00072 *
00073 *  =====================================================================
00074 *
00075 *     .. Parameters ..
00076       REAL               ZERO, ONE
00077       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00078       REAL               ROGUE
00079       PARAMETER          ( ROGUE = -1.0E+10 )
00080 *     ..
00081 *     .. Local Scalars ..
00082       INTEGER            INFO
00083       REAL               ANORM, EPS, RESID
00084 *     ..
00085 *     .. External Functions ..
00086       REAL               SLAMCH, SLANGE, SLANSY
00087       EXTERNAL           SLAMCH, SLANGE, SLANSY
00088 *     ..
00089 *     .. External Subroutines ..
00090       EXTERNAL           SGEMM, SLACPY, SLASET, SORGQR, SSYRK
00091 *     ..
00092 *     .. Intrinsic Functions ..
00093       INTRINSIC          MAX, REAL
00094 *     ..
00095 *     .. Scalars in Common ..
00096       CHARACTER*32       SRNAMT
00097 *     ..
00098 *     .. Common blocks ..
00099       COMMON             / SRNAMC / SRNAMT
00100 *     ..
00101 *     .. Executable Statements ..
00102 *
00103       EPS = SLAMCH( 'Epsilon' )
00104 *
00105 *     Copy the first k columns of the factorization to the array Q
00106 *
00107       CALL SLASET( 'Full', M, N, ROGUE, ROGUE, Q, LDA )
00108       CALL SLACPY( 'Lower', M-1, K, AF( 2, 1 ), LDA, Q( 2, 1 ), LDA )
00109 *
00110 *     Generate the first n columns of the matrix Q
00111 *
00112       SRNAMT = 'SORGQR'
00113       CALL SORGQR( M, N, K, Q, LDA, TAU, WORK, LWORK, INFO )
00114 *
00115 *     Copy R(1:n,1:k)
00116 *
00117       CALL SLASET( 'Full', N, K, ZERO, ZERO, R, LDA )
00118       CALL SLACPY( 'Upper', N, K, AF, LDA, R, LDA )
00119 *
00120 *     Compute R(1:n,1:k) - Q(1:m,1:n)' * A(1:m,1:k)
00121 *
00122       CALL SGEMM( 'Transpose', 'No transpose', N, K, M, -ONE, Q, LDA, A,
00123      \$            LDA, ONE, R, LDA )
00124 *
00125 *     Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
00126 *
00127       ANORM = SLANGE( '1', M, K, A, LDA, RWORK )
00128       RESID = SLANGE( '1', N, K, R, LDA, RWORK )
00129       IF( ANORM.GT.ZERO ) THEN
00130          RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, M ) ) ) / ANORM ) / EPS
00131       ELSE
00132          RESULT( 1 ) = ZERO
00133       END IF
00134 *
00135 *     Compute I - Q'*Q
00136 *
00137       CALL SLASET( 'Full', N, N, ZERO, ONE, R, LDA )
00138       CALL SSYRK( 'Upper', 'Transpose', N, M, -ONE, Q, LDA, ONE, R,
00139      \$            LDA )
00140 *
00141 *     Compute norm( I - Q'*Q ) / ( M * EPS ) .
00142 *
00143       RESID = SLANSY( '1', 'Upper', N, R, LDA, RWORK )
00144 *
00145       RESULT( 2 ) = ( RESID / REAL( MAX( 1, M ) ) ) / EPS
00146 *
00147       RETURN
00148 *
00149 *     End of SQRT02
00150 *
00151       END
```