LAPACK 3.3.0

cgeequ.f

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00001       SUBROUTINE CGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
00002      $                   INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       INTEGER            INFO, LDA, M, N
00011       REAL               AMAX, COLCND, ROWCND
00012 *     ..
00013 *     .. Array Arguments ..
00014       REAL               C( * ), R( * )
00015       COMPLEX            A( LDA, * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  CGEEQU computes row and column scalings intended to equilibrate an
00022 *  M-by-N matrix A and reduce its condition number.  R returns the row
00023 *  scale factors and C the column scale factors, chosen to try to make
00024 *  the largest element in each row and column of the matrix B with
00025 *  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
00026 *
00027 *  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
00028 *  number and BIGNUM = largest safe number.  Use of these scaling
00029 *  factors is not guaranteed to reduce the condition number of A but
00030 *  works well in practice.
00031 *
00032 *  Arguments
00033 *  =========
00034 *
00035 *  M       (input) INTEGER
00036 *          The number of rows of the matrix A.  M >= 0.
00037 *
00038 *  N       (input) INTEGER
00039 *          The number of columns of the matrix A.  N >= 0.
00040 *
00041 *  A       (input) COMPLEX array, dimension (LDA,N)
00042 *          The M-by-N matrix whose equilibration factors are
00043 *          to be computed.
00044 *
00045 *  LDA     (input) INTEGER
00046 *          The leading dimension of the array A.  LDA >= max(1,M).
00047 *
00048 *  R       (output) REAL array, dimension (M)
00049 *          If INFO = 0 or INFO > M, R contains the row scale factors
00050 *          for A.
00051 *
00052 *  C       (output) REAL array, dimension (N)
00053 *          If INFO = 0,  C contains the column scale factors for A.
00054 *
00055 *  ROWCND  (output) REAL
00056 *          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
00057 *          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
00058 *          AMAX is neither too large nor too small, it is not worth
00059 *          scaling by R.
00060 *
00061 *  COLCND  (output) REAL
00062 *          If INFO = 0, COLCND contains the ratio of the smallest
00063 *          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
00064 *          worth scaling by C.
00065 *
00066 *  AMAX    (output) REAL
00067 *          Absolute value of largest matrix element.  If AMAX is very
00068 *          close to overflow or very close to underflow, the matrix
00069 *          should be scaled.
00070 *
00071 *  INFO    (output) INTEGER
00072 *          = 0:  successful exit
00073 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00074 *          > 0:  if INFO = i,  and i is
00075 *                <= M:  the i-th row of A is exactly zero
00076 *                >  M:  the (i-M)-th column of A is exactly zero
00077 *
00078 *  =====================================================================
00079 *
00080 *     .. Parameters ..
00081       REAL               ONE, ZERO
00082       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00083 *     ..
00084 *     .. Local Scalars ..
00085       INTEGER            I, J
00086       REAL               BIGNUM, RCMAX, RCMIN, SMLNUM
00087       COMPLEX            ZDUM
00088 *     ..
00089 *     .. External Functions ..
00090       REAL               SLAMCH
00091       EXTERNAL           SLAMCH
00092 *     ..
00093 *     .. External Subroutines ..
00094       EXTERNAL           XERBLA
00095 *     ..
00096 *     .. Intrinsic Functions ..
00097       INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
00098 *     ..
00099 *     .. Statement Functions ..
00100       REAL               CABS1
00101 *     ..
00102 *     .. Statement Function definitions ..
00103       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00104 *     ..
00105 *     .. Executable Statements ..
00106 *
00107 *     Test the input parameters.
00108 *
00109       INFO = 0
00110       IF( M.LT.0 ) THEN
00111          INFO = -1
00112       ELSE IF( N.LT.0 ) THEN
00113          INFO = -2
00114       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00115          INFO = -4
00116       END IF
00117       IF( INFO.NE.0 ) THEN
00118          CALL XERBLA( 'CGEEQU', -INFO )
00119          RETURN
00120       END IF
00121 *
00122 *     Quick return if possible
00123 *
00124       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00125          ROWCND = ONE
00126          COLCND = ONE
00127          AMAX = ZERO
00128          RETURN
00129       END IF
00130 *
00131 *     Get machine constants.
00132 *
00133       SMLNUM = SLAMCH( 'S' )
00134       BIGNUM = ONE / SMLNUM
00135 *
00136 *     Compute row scale factors.
00137 *
00138       DO 10 I = 1, M
00139          R( I ) = ZERO
00140    10 CONTINUE
00141 *
00142 *     Find the maximum element in each row.
00143 *
00144       DO 30 J = 1, N
00145          DO 20 I = 1, M
00146             R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
00147    20    CONTINUE
00148    30 CONTINUE
00149 *
00150 *     Find the maximum and minimum scale factors.
00151 *
00152       RCMIN = BIGNUM
00153       RCMAX = ZERO
00154       DO 40 I = 1, M
00155          RCMAX = MAX( RCMAX, R( I ) )
00156          RCMIN = MIN( RCMIN, R( I ) )
00157    40 CONTINUE
00158       AMAX = RCMAX
00159 *
00160       IF( RCMIN.EQ.ZERO ) THEN
00161 *
00162 *        Find the first zero scale factor and return an error code.
00163 *
00164          DO 50 I = 1, M
00165             IF( R( I ).EQ.ZERO ) THEN
00166                INFO = I
00167                RETURN
00168             END IF
00169    50    CONTINUE
00170       ELSE
00171 *
00172 *        Invert the scale factors.
00173 *
00174          DO 60 I = 1, M
00175             R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
00176    60    CONTINUE
00177 *
00178 *        Compute ROWCND = min(R(I)) / max(R(I))
00179 *
00180          ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00181       END IF
00182 *
00183 *     Compute column scale factors
00184 *
00185       DO 70 J = 1, N
00186          C( J ) = ZERO
00187    70 CONTINUE
00188 *
00189 *     Find the maximum element in each column,
00190 *     assuming the row scaling computed above.
00191 *
00192       DO 90 J = 1, N
00193          DO 80 I = 1, M
00194             C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
00195    80    CONTINUE
00196    90 CONTINUE
00197 *
00198 *     Find the maximum and minimum scale factors.
00199 *
00200       RCMIN = BIGNUM
00201       RCMAX = ZERO
00202       DO 100 J = 1, N
00203          RCMIN = MIN( RCMIN, C( J ) )
00204          RCMAX = MAX( RCMAX, C( J ) )
00205   100 CONTINUE
00206 *
00207       IF( RCMIN.EQ.ZERO ) THEN
00208 *
00209 *        Find the first zero scale factor and return an error code.
00210 *
00211          DO 110 J = 1, N
00212             IF( C( J ).EQ.ZERO ) THEN
00213                INFO = M + J
00214                RETURN
00215             END IF
00216   110    CONTINUE
00217       ELSE
00218 *
00219 *        Invert the scale factors.
00220 *
00221          DO 120 J = 1, N
00222             C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
00223   120    CONTINUE
00224 *
00225 *        Compute COLCND = min(C(J)) / max(C(J))
00226 *
00227          COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00228       END IF
00229 *
00230       RETURN
00231 *
00232 *     End of CGEEQU
00233 *
00234       END
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