Annotation of rpl/lapack/lapack/zheequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2.2) --
! 4: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
! 5: * -- Jason Riedy of Univ. of California Berkeley. --
! 6: * -- June 2010 --
! 7: *
! 8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 9: * -- Univ. of California Berkeley and NAG Ltd. --
! 10: *
! 11: IMPLICIT NONE
! 12: * ..
! 13: * .. Scalar Arguments ..
! 14: INTEGER INFO, LDA, N
! 15: DOUBLE PRECISION AMAX, SCOND
! 16: CHARACTER UPLO
! 17: * ..
! 18: * .. Array Arguments ..
! 19: COMPLEX*16 A( LDA, * ), WORK( * )
! 20: DOUBLE PRECISION S( * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * ZSYEQUB computes row and column scalings intended to equilibrate a
! 27: * symmetric matrix A and reduce its condition number
! 28: * (with respect to the two-norm). S contains the scale factors,
! 29: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
! 30: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
! 31: * choice of S puts the condition number of B within a factor N of the
! 32: * smallest possible condition number over all possible diagonal
! 33: * scalings.
! 34: *
! 35: * Arguments
! 36: * =========
! 37: *
! 38: * N (input) INTEGER
! 39: * The order of the matrix A. N >= 0.
! 40: *
! 41: * A (input) COMPLEX*16 array, dimension (LDA,N)
! 42: * The N-by-N symmetric matrix whose scaling
! 43: * factors are to be computed. Only the diagonal elements of A
! 44: * are referenced.
! 45: *
! 46: * LDA (input) INTEGER
! 47: * The leading dimension of the array A. LDA >= max(1,N).
! 48: *
! 49: * S (output) DOUBLE PRECISION array, dimension (N)
! 50: * If INFO = 0, S contains the scale factors for A.
! 51: *
! 52: * SCOND (output) DOUBLE PRECISION
! 53: * If INFO = 0, S contains the ratio of the smallest S(i) to
! 54: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 55: * large nor too small, it is not worth scaling by S.
! 56: *
! 57: * AMAX (output) DOUBLE PRECISION
! 58: * Absolute value of largest matrix element. If AMAX is very
! 59: * close to overflow or very close to underflow, the matrix
! 60: * should be scaled.
! 61: * INFO (output) INTEGER
! 62: * = 0: successful exit
! 63: * < 0: if INFO = -i, the i-th argument had an illegal value
! 64: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 65: *
! 66: * =====================================================================
! 67: *
! 68: * .. Parameters ..
! 69: DOUBLE PRECISION ONE, ZERO
! 70: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 71: INTEGER MAX_ITER
! 72: PARAMETER ( MAX_ITER = 100 )
! 73: * ..
! 74: * .. Local Scalars ..
! 75: INTEGER I, J, ITER
! 76: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D,
! 77: $ BASE, SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
! 78: LOGICAL UP
! 79: COMPLEX*16 ZDUM
! 80: * ..
! 81: * .. External Functions ..
! 82: DOUBLE PRECISION DLAMCH
! 83: LOGICAL LSAME
! 84: EXTERNAL DLAMCH, LSAME
! 85: * ..
! 86: * .. External Subroutines ..
! 87: EXTERNAL ZLASSQ
! 88: * ..
! 89: * .. Intrinsic Functions ..
! 90: INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
! 91: * ..
! 92: * .. Statement Functions ..
! 93: DOUBLE PRECISION CABS1
! 94: * ..
! 95: * .. Statement Function Definitions ..
! 96: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 97: *
! 98: * Test input parameters.
! 99: *
! 100: INFO = 0
! 101: IF (.NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
! 102: INFO = -1
! 103: ELSE IF ( N .LT. 0 ) THEN
! 104: INFO = -2
! 105: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
! 106: INFO = -4
! 107: END IF
! 108: IF ( INFO .NE. 0 ) THEN
! 109: CALL XERBLA( 'ZHEEQUB', -INFO )
! 110: RETURN
! 111: END IF
! 112:
! 113: UP = LSAME( UPLO, 'U' )
! 114: AMAX = ZERO
! 115: *
! 116: * Quick return if possible.
! 117: *
! 118: IF ( N .EQ. 0 ) THEN
! 119: SCOND = ONE
! 120: RETURN
! 121: END IF
! 122:
! 123: DO I = 1, N
! 124: S( I ) = ZERO
! 125: END DO
! 126:
! 127: AMAX = ZERO
! 128: IF ( UP ) THEN
! 129: DO J = 1, N
! 130: DO I = 1, J-1
! 131: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
! 132: S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
! 133: AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
! 134: END DO
! 135: S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
! 136: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
! 137: END DO
! 138: ELSE
! 139: DO J = 1, N
! 140: S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
! 141: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
! 142: DO I = J+1, N
! 143: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
! 144: S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
! 145: AMAX = MAX( AMAX, CABS1( A(I, J ) ) )
! 146: END DO
! 147: END DO
! 148: END IF
! 149: DO J = 1, N
! 150: S( J ) = 1.0D+0 / S( J )
! 151: END DO
! 152:
! 153: TOL = ONE / SQRT( 2.0D0 * N )
! 154:
! 155: DO ITER = 1, MAX_ITER
! 156: SCALE = 0.0D+0
! 157: SUMSQ = 0.0D+0
! 158: * beta = |A|s
! 159: DO I = 1, N
! 160: WORK( I ) = ZERO
! 161: END DO
! 162: IF ( UP ) THEN
! 163: DO J = 1, N
! 164: DO I = 1, J-1
! 165: T = CABS1( A( I, J ) )
! 166: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
! 167: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
! 168: END DO
! 169: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
! 170: END DO
! 171: ELSE
! 172: DO J = 1, N
! 173: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
! 174: DO I = J+1, N
! 175: T = CABS1( A( I, J ) )
! 176: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
! 177: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
! 178: END DO
! 179: END DO
! 180: END IF
! 181:
! 182: * avg = s^T beta / n
! 183: AVG = 0.0D+0
! 184: DO I = 1, N
! 185: AVG = AVG + S( I )*WORK( I )
! 186: END DO
! 187: AVG = AVG / N
! 188:
! 189: STD = 0.0D+0
! 190: DO I = 2*N+1, 3*N
! 191: WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
! 192: END DO
! 193: CALL ZLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
! 194: STD = SCALE * SQRT( SUMSQ / N )
! 195:
! 196: IF ( STD .LT. TOL * AVG ) GOTO 999
! 197:
! 198: DO I = 1, N
! 199: T = CABS1( A( I, I ) )
! 200: SI = S( I )
! 201: C2 = ( N-1 ) * T
! 202: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
! 203: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
! 204:
! 205: D = C1*C1 - 4*C0*C2
! 206: IF ( D .LE. 0 ) THEN
! 207: INFO = -1
! 208: RETURN
! 209: END IF
! 210: SI = -2*C0 / ( C1 + SQRT( D ) )
! 211:
! 212: D = SI - S(I)
! 213: U = ZERO
! 214: IF ( UP ) THEN
! 215: DO J = 1, I
! 216: T = CABS1( A( J, I ) )
! 217: U = U + S( J )*T
! 218: WORK( J ) = WORK( J ) + D*T
! 219: END DO
! 220: DO J = I+1,N
! 221: T = CABS1( A( I, J ) )
! 222: U = U + S( J )*T
! 223: WORK( J ) = WORK( J ) + D*T
! 224: END DO
! 225: ELSE
! 226: DO J = 1, I
! 227: T = CABS1( A( I, J ) )
! 228: U = U + S( J )*T
! 229: WORK( J ) = WORK( J ) + D*T
! 230: END DO
! 231: DO J = I+1,N
! 232: T = CABS1( A( J, I ) )
! 233: U = U + S( J )*T
! 234: WORK( J ) = WORK( J ) + D*T
! 235: END DO
! 236: END IF
! 237: AVG = AVG + ( U + WORK( I ) ) * D / N
! 238: S( I ) = SI
! 239: END DO
! 240:
! 241: END DO
! 242:
! 243: 999 CONTINUE
! 244:
! 245: SMLNUM = DLAMCH( 'SAFEMIN' )
! 246: BIGNUM = ONE / SMLNUM
! 247: SMIN = BIGNUM
! 248: SMAX = ZERO
! 249: T = ONE / SQRT( AVG )
! 250: BASE = DLAMCH( 'B' )
! 251: U = ONE / LOG( BASE )
! 252: DO I = 1, N
! 253: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
! 254: SMIN = MIN( SMIN, S( I ) )
! 255: SMAX = MAX( SMAX, S( I ) )
! 256: END DO
! 257: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 258:
! 259: END
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