LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsyequb.f
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00001 *> \brief \b DSYEQUB
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download DSYEQUB + dependencies 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DSYEQUB( 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 *       DOUBLE PRECISION   A( LDA, * ), S( * ), WORK( * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> DSYEQUB computes row and column scalings intended to equilibrate a
00039 *> symmetric matrix A and reduce its condition number
00040 *> (with respect to the two-norm).  S contains the scale factors,
00041 *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
00042 *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
00043 *> choice of S puts the condition number of B within a factor N of the
00044 *> smallest possible condition number over all possible diagonal
00045 *> scalings.
00046 *> \endverbatim
00047 *
00048 *  Arguments:
00049 *  ==========
00050 *
00051 *> \param[in] UPLO
00052 *> \verbatim
00053 *>          UPLO is CHARACTER*1
00054 *>          Specifies whether the details of the factorization are stored
00055 *>          as an upper or lower triangular matrix.
00056 *>          = 'U':  Upper triangular, form is A = U*D*U**T;
00057 *>          = 'L':  Lower triangular, form is A = L*D*L**T.
00058 *> \endverbatim
00059 *>
00060 *> \param[in] N
00061 *> \verbatim
00062 *>          N is INTEGER
00063 *>          The order of the matrix A.  N >= 0.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] A
00067 *> \verbatim
00068 *>          A is DOUBLE PRECISION array, dimension (LDA,N)
00069 *>          The N-by-N symmetric matrix whose scaling
00070 *>          factors are to be computed.  Only the diagonal elements of A
00071 *>          are referenced.
00072 *> \endverbatim
00073 *>
00074 *> \param[in] LDA
00075 *> \verbatim
00076 *>          LDA is INTEGER
00077 *>          The leading dimension of the array A.  LDA >= max(1,N).
00078 *> \endverbatim
00079 *>
00080 *> \param[out] S
00081 *> \verbatim
00082 *>          S is DOUBLE PRECISION array, dimension (N)
00083 *>          If INFO = 0, S contains the scale factors for A.
00084 *> \endverbatim
00085 *>
00086 *> \param[out] SCOND
00087 *> \verbatim
00088 *>          SCOND is DOUBLE PRECISION
00089 *>          If INFO = 0, S contains the ratio of the smallest S(i) to
00090 *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00091 *>          large nor too small, it is not worth scaling by S.
00092 *> \endverbatim
00093 *>
00094 *> \param[out] AMAX
00095 *> \verbatim
00096 *>          AMAX is DOUBLE PRECISION
00097 *>          Absolute value of largest matrix element.  If AMAX is very
00098 *>          close to overflow or very close to underflow, the matrix
00099 *>          should be scaled.
00100 *> \endverbatim
00101 *>
00102 *> \param[out] WORK
00103 *> \verbatim
00104 *>          WORK is DOUBLE PRECISION array, dimension (3*N)
00105 *> \endverbatim
00106 *>
00107 *> \param[out] INFO
00108 *> \verbatim
00109 *>          INFO is INTEGER
00110 *>          = 0:  successful exit
00111 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00112 *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00113 *> \endverbatim
00114 *
00115 *  Authors:
00116 *  ========
00117 *
00118 *> \author Univ. of Tennessee 
00119 *> \author Univ. of California Berkeley 
00120 *> \author Univ. of Colorado Denver 
00121 *> \author NAG Ltd. 
00122 *
00123 *> \date November 2011
00124 *
00125 *> \ingroup doubleSYcomputational
00126 *
00127 *> \par References:
00128 *  ================
00129 *>
00130 *>  Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
00131 *>  Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
00132 *>  DOI 10.1023/B:NUMA.0000016606.32820.69 \n
00133 *>  Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
00134 *>
00135 *  =====================================================================
00136       SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
00137 *
00138 *  -- LAPACK computational routine (version 3.4.0) --
00139 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00140 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00141 *     November 2011
00142 *
00143 *     .. Scalar Arguments ..
00144       INTEGER            INFO, LDA, N
00145       DOUBLE PRECISION   AMAX, SCOND
00146       CHARACTER          UPLO
00147 *     ..
00148 *     .. Array Arguments ..
00149       DOUBLE PRECISION   A( LDA, * ), S( * ), WORK( * )
00150 *     ..
00151 *
00152 *  =====================================================================
00153 *
00154 *     .. Parameters ..
00155       DOUBLE PRECISION   ONE, ZERO
00156       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00157       INTEGER            MAX_ITER
00158       PARAMETER          ( MAX_ITER = 100 )
00159 *     ..
00160 *     .. Local Scalars ..
00161       INTEGER            I, J, ITER
00162       DOUBLE PRECISION   AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
00163      $                   SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
00164       LOGICAL            UP
00165 *     ..
00166 *     .. External Functions ..
00167       DOUBLE PRECISION   DLAMCH
00168       LOGICAL            LSAME
00169       EXTERNAL           DLAMCH, LSAME
00170 *     ..
00171 *     .. External Subroutines ..
00172       EXTERNAL           DLASSQ
00173 *     ..
00174 *     .. Intrinsic Functions ..
00175       INTRINSIC          ABS, INT, LOG, MAX, MIN, SQRT
00176 *     ..
00177 *     .. Executable Statements ..
00178 *
00179 *     Test input parameters.
00180 *
00181       INFO = 0
00182       IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
00183         INFO = -1
00184       ELSE IF ( N .LT. 0 ) THEN
00185         INFO = -2
00186       ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
00187         INFO = -4
00188       END IF
00189       IF ( INFO .NE. 0 ) THEN
00190         CALL XERBLA( 'DSYEQUB', -INFO )
00191         RETURN
00192       END IF
00193 
00194       UP = LSAME( UPLO, 'U' )
00195       AMAX = ZERO
00196 *
00197 *     Quick return if possible.
00198 *
00199       IF ( N .EQ. 0 ) THEN
00200         SCOND = ONE
00201         RETURN
00202       END IF
00203 
00204       DO I = 1, N
00205         S( I ) = ZERO
00206       END DO
00207 
00208       AMAX = ZERO
00209       IF ( UP ) THEN
00210          DO J = 1, N
00211             DO I = 1, J-1
00212                S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
00213                S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
00214                AMAX = MAX( AMAX, ABS( A(I, J) ) )
00215             END DO
00216             S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
00217             AMAX = MAX( AMAX, ABS( A( J, J ) ) )
00218          END DO
00219       ELSE
00220          DO J = 1, N
00221             S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
00222             AMAX = MAX( AMAX, ABS( A( J, J ) ) )
00223             DO I = J+1, N
00224                S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
00225                S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
00226                AMAX = MAX( AMAX, ABS( A( I, J ) ) )
00227             END DO
00228          END DO
00229       END IF
00230       DO J = 1, N
00231          S( J ) = 1.0D+0 / S( J )
00232       END DO
00233 
00234       TOL = ONE / SQRT(2.0D0 * N)
00235 
00236       DO ITER = 1, MAX_ITER
00237          SCALE = 0.0D+0
00238          SUMSQ = 0.0D+0
00239 *       BETA = |A|S
00240         DO I = 1, N
00241            WORK(I) = ZERO
00242         END DO
00243         IF ( UP ) THEN
00244            DO J = 1, N
00245               DO I = 1, J-1
00246                  T = ABS( A( I, J ) )
00247                  WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
00248                  WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
00249               END DO
00250               WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
00251            END DO
00252         ELSE
00253            DO J = 1, N
00254               WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
00255               DO I = J+1, N
00256                  T = ABS( A( I, J ) )
00257                  WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
00258                  WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
00259               END DO
00260            END DO
00261         END IF
00262 
00263 *       avg = s^T beta / n
00264         AVG = 0.0D+0
00265         DO I = 1, N
00266           AVG = AVG + S( I )*WORK( I )
00267         END DO
00268         AVG = AVG / N
00269 
00270         STD = 0.0D+0
00271         DO I = 2*N+1, 3*N
00272            WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
00273         END DO
00274         CALL DLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
00275         STD = SCALE * SQRT( SUMSQ / N )
00276 
00277         IF ( STD .LT. TOL * AVG ) GOTO 999
00278 
00279         DO I = 1, N
00280           T = ABS( A( I, I ) )
00281           SI = S( I )
00282           C2 = ( N-1 ) * T
00283           C1 = ( N-2 ) * ( WORK( I ) - T*SI )
00284           C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
00285           D = C1*C1 - 4*C0*C2
00286 
00287           IF ( D .LE. 0 ) THEN
00288             INFO = -1
00289             RETURN
00290           END IF
00291           SI = -2*C0 / ( C1 + SQRT( D ) )
00292 
00293           D = SI - S( I )
00294           U = ZERO
00295           IF ( UP ) THEN
00296             DO J = 1, I
00297               T = ABS( A( J, I ) )
00298               U = U + S( J )*T
00299               WORK( J ) = WORK( J ) + D*T
00300             END DO
00301             DO J = I+1,N
00302               T = ABS( A( I, J ) )
00303               U = U + S( J )*T
00304               WORK( J ) = WORK( J ) + D*T
00305             END DO
00306           ELSE
00307             DO J = 1, I
00308               T = ABS( A( I, J ) )
00309               U = U + S( J )*T
00310               WORK( J ) = WORK( J ) + D*T
00311             END DO
00312             DO J = I+1,N
00313               T = ABS( A( J, I ) )
00314               U = U + S( J )*T
00315               WORK( J ) = WORK( J ) + D*T
00316             END DO
00317           END IF
00318 
00319           AVG = AVG + ( U + WORK( I ) ) * D / N
00320           S( I ) = SI
00321 
00322         END DO
00323 
00324       END DO
00325 
00326  999  CONTINUE
00327 
00328       SMLNUM = DLAMCH( 'SAFEMIN' )
00329       BIGNUM = ONE / SMLNUM
00330       SMIN = BIGNUM
00331       SMAX = ZERO
00332       T = ONE / SQRT(AVG)
00333       BASE = DLAMCH( 'B' )
00334       U = ONE / LOG( BASE )
00335       DO I = 1, N
00336         S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
00337         SMIN = MIN( SMIN, S( I ) )
00338         SMAX = MAX( SMAX, S( I ) )
00339       END DO
00340       SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
00341 *
00342       END
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