LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zheequb.f
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00001 *> \brief \b ZHEEQUB
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       INTEGER            INFO, LDA, N
00025 *       DOUBLE PRECISION   AMAX, SCOND
00026 *       CHARACTER          UPLO
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       COMPLEX*16         A( LDA, * ), WORK( * )
00030 *       DOUBLE PRECISION   S( * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *> ZHEEQUB computes row and column scalings intended to equilibrate a
00040 *> Hermitian matrix A and reduce its condition number
00041 *> (with respect to the two-norm).  S contains the scale factors,
00042 *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
00043 *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
00044 *> choice of S puts the condition number of B within a factor N of the
00045 *> smallest possible condition number over all possible diagonal
00046 *> scalings.
00047 *> \endverbatim
00048 *
00049 *  Arguments:
00050 *  ==========
00051 *
00052 *> \param[in] UPLO
00053 *> \verbatim
00054 *>          UPLO is CHARACTER*1
00055 *>          = 'U':  Upper triangles of A and B are stored;
00056 *>          = 'L':  Lower triangles of A and B are stored.
00057 *> \endverbatim
00058 *>
00059 *> \param[in] N
00060 *> \verbatim
00061 *>          N is INTEGER
00062 *>          The order of the matrix A.  N >= 0.
00063 *> \endverbatim
00064 *>
00065 *> \param[in] A
00066 *> \verbatim
00067 *>          A is COMPLEX*16 array, dimension (LDA,N)
00068 *>          The N-by-N Hermitian matrix whose scaling
00069 *>          factors are to be computed.  Only the diagonal elements of A
00070 *>          are referenced.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] LDA
00074 *> \verbatim
00075 *>          LDA is INTEGER
00076 *>          The leading dimension of the array A.  LDA >= max(1,N).
00077 *> \endverbatim
00078 *>
00079 *> \param[out] S
00080 *> \verbatim
00081 *>          S is DOUBLE PRECISION array, dimension (N)
00082 *>          If INFO = 0, S contains the scale factors for A.
00083 *> \endverbatim
00084 *>
00085 *> \param[out] SCOND
00086 *> \verbatim
00087 *>          SCOND is DOUBLE PRECISION
00088 *>          If INFO = 0, S contains the ratio of the smallest S(i) to
00089 *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00090 *>          large nor too small, it is not worth scaling by S.
00091 *> \endverbatim
00092 *>
00093 *> \param[out] AMAX
00094 *> \verbatim
00095 *>          AMAX is DOUBLE PRECISION
00096 *>          Absolute value of largest matrix element.  If AMAX is very
00097 *>          close to overflow or very close to underflow, the matrix
00098 *>          should be scaled.
00099 *> \endverbatim
00100 *>
00101 *> \param[out] WORK
00102 *> \verbatim
00103 *>          WORK is DOUBLE PRECISION array, dimension (3*N)
00104 *> \endverbatim
00105 *>
00106 *> \param[out] INFO
00107 *> \verbatim
00108 *>          INFO is INTEGER
00109 *>          = 0:  successful exit
00110 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00111 *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00112 *> \endverbatim
00113 *
00114 *  Authors:
00115 *  ========
00116 *
00117 *> \author Univ. of Tennessee 
00118 *> \author Univ. of California Berkeley 
00119 *> \author Univ. of Colorado Denver 
00120 *> \author NAG Ltd. 
00121 *
00122 *> \date April 2012
00123 *
00124 *> \ingroup complex16HEcomputational
00125 *
00126 *  =====================================================================
00127       SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
00128 *
00129 *  -- LAPACK computational routine (version 3.4.1) --
00130 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00131 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00132 *     April 2012
00133 *
00134 *     .. Scalar Arguments ..
00135       INTEGER            INFO, LDA, N
00136       DOUBLE PRECISION   AMAX, SCOND
00137       CHARACTER          UPLO
00138 *     ..
00139 *     .. Array Arguments ..
00140       COMPLEX*16         A( LDA, * ), WORK( * )
00141       DOUBLE PRECISION   S( * )
00142 *     ..
00143 *
00144 *  =====================================================================
00145 *
00146 *     .. Parameters ..
00147       DOUBLE PRECISION   ONE, ZERO
00148       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00149       INTEGER            MAX_ITER
00150       PARAMETER          ( MAX_ITER = 100 )
00151 *     ..
00152 *     .. Local Scalars ..
00153       INTEGER            I, J, ITER
00154       DOUBLE PRECISION   AVG, STD, TOL, C0, C1, C2, T, U, SI, D,
00155      $                   BASE, SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
00156       LOGICAL            UP
00157       COMPLEX*16         ZDUM
00158 *     ..
00159 *     .. External Functions ..
00160       DOUBLE PRECISION   DLAMCH
00161       LOGICAL            LSAME
00162       EXTERNAL           DLAMCH, LSAME
00163 *     ..
00164 *     .. External Subroutines ..
00165       EXTERNAL           ZLASSQ
00166 *     ..
00167 *     .. Intrinsic Functions ..
00168       INTRINSIC          ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
00169 *     ..
00170 *     .. Statement Functions ..
00171       DOUBLE PRECISION   CABS1
00172 *     ..
00173 *     .. Statement Function Definitions ..
00174       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00175 *
00176 *     Test input parameters.
00177 *
00178       INFO = 0
00179       IF (.NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
00180         INFO = -1
00181       ELSE IF ( N .LT. 0 ) THEN
00182         INFO = -2
00183       ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
00184         INFO = -4
00185       END IF
00186       IF ( INFO .NE. 0 ) THEN
00187         CALL XERBLA( 'ZHEEQUB', -INFO )
00188         RETURN
00189       END IF
00190 
00191       UP = LSAME( UPLO, 'U' )
00192       AMAX = ZERO
00193 *
00194 *     Quick return if possible.
00195 *
00196       IF ( N .EQ. 0 ) THEN
00197         SCOND = ONE
00198         RETURN
00199       END IF
00200 
00201       DO I = 1, N
00202         S( I ) = ZERO
00203       END DO
00204 
00205       AMAX = ZERO
00206       IF ( UP ) THEN
00207          DO J = 1, N
00208             DO I = 1, J-1
00209                S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
00210                S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
00211                AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
00212             END DO
00213             S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
00214             AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
00215          END DO
00216       ELSE
00217          DO J = 1, N
00218             S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
00219             AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
00220             DO I = J+1, N
00221                S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
00222                S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
00223                AMAX = MAX( AMAX, CABS1( A(I, J ) ) )
00224             END DO
00225          END DO
00226       END IF
00227       DO J = 1, N
00228          S( J ) = 1.0D+0 / S( J )
00229       END DO
00230 
00231       TOL = ONE / SQRT( 2.0D0 * N )
00232 
00233       DO ITER = 1, MAX_ITER
00234          SCALE = 0.0D+0
00235          SUMSQ = 0.0D+0
00236 *       beta = |A|s
00237         DO I = 1, N
00238            WORK( I ) = ZERO
00239         END DO
00240         IF ( UP ) THEN
00241            DO J = 1, N
00242               DO I = 1, J-1
00243                  T = CABS1( A( I, J ) )
00244                  WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
00245                  WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
00246               END DO
00247               WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
00248            END DO
00249         ELSE
00250            DO J = 1, N
00251               WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
00252               DO I = J+1, N
00253                  T = CABS1( A( I, J ) )
00254                  WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
00255                  WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
00256               END DO
00257            END DO
00258         END IF
00259 
00260 *       avg = s^T beta / n
00261         AVG = 0.0D+0
00262         DO I = 1, N
00263           AVG = AVG + S( I )*WORK( I )
00264         END DO
00265         AVG = AVG / N
00266 
00267         STD = 0.0D+0
00268         DO I = 2*N+1, 3*N
00269            WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
00270         END DO
00271         CALL ZLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
00272         STD = SCALE * SQRT( SUMSQ / N )
00273 
00274         IF ( STD .LT. TOL * AVG ) GOTO 999
00275 
00276         DO I = 1, N
00277           T = CABS1( A( I, I ) )
00278           SI = S( I )
00279           C2 = ( N-1 ) * T
00280           C1 = ( N-2 ) * ( WORK( I ) - T*SI )
00281           C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
00282 
00283           D = C1*C1 - 4*C0*C2
00284           IF ( D .LE. 0 ) THEN
00285             INFO = -1
00286             RETURN
00287           END IF
00288           SI = -2*C0 / ( C1 + SQRT( D ) )
00289 
00290           D = SI - S(I)
00291           U = ZERO
00292           IF ( UP ) THEN
00293             DO J = 1, I
00294               T = CABS1( A( J, I ) )
00295               U = U + S( J )*T
00296               WORK( J ) = WORK( J ) + D*T
00297             END DO
00298             DO J = I+1,N
00299               T = CABS1( A( I, J ) )
00300               U = U + S( J )*T
00301               WORK( J ) = WORK( J ) + D*T
00302             END DO
00303           ELSE
00304             DO J = 1, I
00305               T = CABS1( A( I, J ) )
00306               U = U + S( J )*T
00307               WORK( J ) = WORK( J ) + D*T
00308             END DO
00309             DO J = I+1,N
00310               T = CABS1( A( J, I ) )
00311               U = U + S( J )*T
00312               WORK( J ) = WORK( J ) + D*T
00313             END DO
00314           END IF
00315           AVG = AVG + ( U + WORK( I ) ) * D / N
00316           S( I ) = SI
00317         END DO
00318 
00319       END DO
00320 
00321  999  CONTINUE
00322 
00323       SMLNUM = DLAMCH( 'SAFEMIN' )
00324       BIGNUM = ONE / SMLNUM
00325       SMIN = BIGNUM
00326       SMAX = ZERO
00327       T = ONE / SQRT( AVG )
00328       BASE = DLAMCH( 'B' )
00329       U = ONE / LOG( BASE )
00330       DO I = 1, N
00331         S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
00332         SMIN = MIN( SMIN, S( I ) )
00333         SMAX = MAX( SMAX, S( I ) )
00334       END DO
00335       SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
00336 
00337       END
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