Annotation of rpl/lapack/lapack/zsyequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZSYEQUB( 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: * UPLO (input) CHARACTER*1
! 39: * Specifies whether the details of the factorization are stored
! 40: * as an upper or lower triangular matrix.
! 41: * = 'U': Upper triangular, form is A = U*D*U**T;
! 42: * = 'L': Lower triangular, form is A = L*D*L**T.
! 43: *
! 44: * N (input) INTEGER
! 45: * The order of the matrix A. N >= 0.
! 46: *
! 47: * A (input) COMPLEX*16 array, dimension (LDA,N)
! 48: * The N-by-N symmetric matrix whose scaling
! 49: * factors are to be computed. Only the diagonal elements of A
! 50: * are referenced.
! 51: *
! 52: * LDA (input) INTEGER
! 53: * The leading dimension of the array A. LDA >= max(1,N).
! 54: *
! 55: * S (output) DOUBLE PRECISION array, dimension (N)
! 56: * If INFO = 0, S contains the scale factors for A.
! 57: *
! 58: * SCOND (output) DOUBLE PRECISION
! 59: * If INFO = 0, S contains the ratio of the smallest S(i) to
! 60: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 61: * large nor too small, it is not worth scaling by S.
! 62: *
! 63: * AMAX (output) DOUBLE PRECISION
! 64: * Absolute value of largest matrix element. If AMAX is very
! 65: * close to overflow or very close to underflow, the matrix
! 66: * should be scaled.
! 67: *
! 68: * WORK (workspace) COMPLEX*16 array, dimension (3*N)
! 69: *
! 70: * INFO (output) INTEGER
! 71: * = 0: successful exit
! 72: * < 0: if INFO = -i, the i-th argument had an illegal value
! 73: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 74: *
! 75: * Further Details
! 76: * ======= =======
! 77: *
! 78: * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
! 79: * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
! 80: * DOI 10.1023/B:NUMA.0000016606.32820.69
! 81: * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
! 82: *
! 83: * =====================================================================
! 84: *
! 85: * .. Parameters ..
! 86: DOUBLE PRECISION ONE, ZERO
! 87: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
! 88: INTEGER MAX_ITER
! 89: PARAMETER ( MAX_ITER = 100 )
! 90: * ..
! 91: * .. Local Scalars ..
! 92: INTEGER I, J, ITER
! 93: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
! 94: $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
! 95: LOGICAL UP
! 96: COMPLEX*16 ZDUM
! 97: * ..
! 98: * .. External Functions ..
! 99: DOUBLE PRECISION DLAMCH
! 100: LOGICAL LSAME
! 101: EXTERNAL DLAMCH, LSAME
! 102: * ..
! 103: * .. External Subroutines ..
! 104: EXTERNAL ZLASSQ
! 105: * ..
! 106: * .. Intrinsic Functions ..
! 107: INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
! 108: * ..
! 109: * .. Statement Functions ..
! 110: DOUBLE PRECISION CABS1
! 111: * ..
! 112: * Statement Function Definitions
! 113: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 114: * ..
! 115: * .. Executable Statements ..
! 116: *
! 117: * Test the input parameters.
! 118: *
! 119: INFO = 0
! 120: IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
! 121: INFO = -1
! 122: ELSE IF ( N .LT. 0 ) THEN
! 123: INFO = -2
! 124: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
! 125: INFO = -4
! 126: END IF
! 127: IF ( INFO .NE. 0 ) THEN
! 128: CALL XERBLA( 'ZSYEQUB', -INFO )
! 129: RETURN
! 130: END IF
! 131:
! 132: UP = LSAME( UPLO, 'U' )
! 133: AMAX = ZERO
! 134: *
! 135: * Quick return if possible.
! 136: *
! 137: IF ( N .EQ. 0 ) THEN
! 138: SCOND = ONE
! 139: RETURN
! 140: END IF
! 141:
! 142: DO I = 1, N
! 143: S( I ) = ZERO
! 144: END DO
! 145:
! 146: AMAX = ZERO
! 147: IF ( UP ) THEN
! 148: DO J = 1, N
! 149: DO I = 1, J-1
! 150: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
! 151: S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
! 152: AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
! 153: END DO
! 154: S( J ) = MAX( S( J ), CABS1( A( J, J) ) )
! 155: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
! 156: END DO
! 157: ELSE
! 158: DO J = 1, N
! 159: S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
! 160: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
! 161: DO I = J+1, N
! 162: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
! 163: S( J ) = MAX( S( J ), CABS1 (A( I, J ) ) )
! 164: AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
! 165: END DO
! 166: END DO
! 167: END IF
! 168: DO J = 1, N
! 169: S( J ) = 1.0D+0 / S( J )
! 170: END DO
! 171:
! 172: TOL = ONE / SQRT( 2.0D0 * N )
! 173:
! 174: DO ITER = 1, MAX_ITER
! 175: SCALE = 0.0D+0
! 176: SUMSQ = 0.0D+0
! 177: * beta = |A|s
! 178: DO I = 1, N
! 179: WORK( I ) = ZERO
! 180: END DO
! 181: IF ( UP ) THEN
! 182: DO J = 1, N
! 183: DO I = 1, J-1
! 184: T = CABS1( A( I, J ) )
! 185: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
! 186: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
! 187: END DO
! 188: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
! 189: END DO
! 190: ELSE
! 191: DO J = 1, N
! 192: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
! 193: DO I = J+1, N
! 194: T = CABS1( A( I, J ) )
! 195: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
! 196: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
! 197: END DO
! 198: END DO
! 199: END IF
! 200:
! 201: * avg = s^T beta / n
! 202: AVG = 0.0D+0
! 203: DO I = 1, N
! 204: AVG = AVG + S( I )*WORK( I )
! 205: END DO
! 206: AVG = AVG / N
! 207:
! 208: STD = 0.0D+0
! 209: DO I = N+1, 2*N
! 210: WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
! 211: END DO
! 212: CALL ZLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
! 213: STD = SCALE * SQRT( SUMSQ / N )
! 214:
! 215: IF ( STD .LT. TOL * AVG ) GOTO 999
! 216:
! 217: DO I = 1, N
! 218: T = CABS1( A( I, I ) )
! 219: SI = S( I )
! 220: C2 = ( N-1 ) * T
! 221: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
! 222: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
! 223: D = C1*C1 - 4*C0*C2
! 224:
! 225: IF ( D .LE. 0 ) THEN
! 226: INFO = -1
! 227: RETURN
! 228: END IF
! 229: SI = -2*C0 / ( C1 + SQRT( D ) )
! 230:
! 231: D = SI - S( I )
! 232: U = ZERO
! 233: IF ( UP ) THEN
! 234: DO J = 1, I
! 235: T = CABS1( A( J, I ) )
! 236: U = U + S( J )*T
! 237: WORK( J ) = WORK( J ) + D*T
! 238: END DO
! 239: DO J = I+1,N
! 240: T = CABS1( A( I, J ) )
! 241: U = U + S( J )*T
! 242: WORK( J ) = WORK( J ) + D*T
! 243: END DO
! 244: ELSE
! 245: DO J = 1, I
! 246: T = CABS1( A( I, J ) )
! 247: U = U + S( J )*T
! 248: WORK( J ) = WORK( J ) + D*T
! 249: END DO
! 250: DO J = I+1,N
! 251: T = CABS1( A( J, I ) )
! 252: U = U + S( J )*T
! 253: WORK( J ) = WORK( J ) + D*T
! 254: END DO
! 255: END IF
! 256: AVG = AVG + ( U + WORK( I ) ) * D / N
! 257: S( I ) = SI
! 258: END DO
! 259: END DO
! 260:
! 261: 999 CONTINUE
! 262:
! 263: SMLNUM = DLAMCH( 'SAFEMIN' )
! 264: BIGNUM = ONE / SMLNUM
! 265: SMIN = BIGNUM
! 266: SMAX = ZERO
! 267: T = ONE / SQRT( AVG )
! 268: BASE = DLAMCH( 'B' )
! 269: U = ONE / LOG( BASE )
! 270: DO I = 1, N
! 271: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
! 272: SMIN = MIN( SMIN, S( I ) )
! 273: SMAX = MAX( SMAX, S( I ) )
! 274: END DO
! 275: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 276: *
! 277: END
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