Annotation of rpl/lapack/lapack/dsyequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSYEQUB( 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: DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DSYEQUB computes row and column scalings intended to equilibrate a
! 26: * symmetric matrix A and reduce its condition number
! 27: * (with respect to the two-norm). S contains the scale factors,
! 28: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
! 29: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
! 30: * choice of S puts the condition number of B within a factor N of the
! 31: * smallest possible condition number over all possible diagonal
! 32: * scalings.
! 33: *
! 34: * Arguments
! 35: * =========
! 36: *
! 37: * UPLO (input) CHARACTER*1
! 38: * Specifies whether the details of the factorization are stored
! 39: * as an upper or lower triangular matrix.
! 40: * = 'U': Upper triangular, form is A = U*D*U**T;
! 41: * = 'L': Lower triangular, form is A = L*D*L**T.
! 42: *
! 43: * N (input) INTEGER
! 44: * The order of the matrix A. N >= 0.
! 45: *
! 46: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 47: * The N-by-N symmetric matrix whose scaling
! 48: * factors are to be computed. Only the diagonal elements of A
! 49: * are referenced.
! 50: *
! 51: * LDA (input) INTEGER
! 52: * The leading dimension of the array A. LDA >= max(1,N).
! 53: *
! 54: * S (output) DOUBLE PRECISION array, dimension (N)
! 55: * If INFO = 0, S contains the scale factors for A.
! 56: *
! 57: * SCOND (output) DOUBLE PRECISION
! 58: * If INFO = 0, S contains the ratio of the smallest S(i) to
! 59: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 60: * large nor too small, it is not worth scaling by S.
! 61: *
! 62: * AMAX (output) DOUBLE PRECISION
! 63: * Absolute value of largest matrix element. If AMAX is very
! 64: * close to overflow or very close to underflow, the matrix
! 65: * should be scaled.
! 66: *
! 67: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 68: *
! 69: * INFO (output) INTEGER
! 70: * = 0: successful exit
! 71: * < 0: if INFO = -i, the i-th argument had an illegal value
! 72: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 73: *
! 74: * Further Details
! 75: * ======= =======
! 76: *
! 77: * Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
! 78: * Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
! 79: * DOI 10.1023/B:NUMA.0000016606.32820.69
! 80: * Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
! 81: *
! 82: * =====================================================================
! 83: *
! 84: * .. Parameters ..
! 85: DOUBLE PRECISION ONE, ZERO
! 86: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 87: INTEGER MAX_ITER
! 88: PARAMETER ( MAX_ITER = 100 )
! 89: * ..
! 90: * .. Local Scalars ..
! 91: INTEGER I, J, ITER
! 92: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
! 93: $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
! 94: LOGICAL UP
! 95: * ..
! 96: * .. External Functions ..
! 97: DOUBLE PRECISION DLAMCH
! 98: LOGICAL LSAME
! 99: EXTERNAL DLAMCH, LSAME
! 100: * ..
! 101: * .. External Subroutines ..
! 102: EXTERNAL DLASSQ
! 103: * ..
! 104: * .. Intrinsic Functions ..
! 105: INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
! 106: * ..
! 107: * .. Executable Statements ..
! 108: *
! 109: * Test input parameters.
! 110: *
! 111: INFO = 0
! 112: IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
! 113: INFO = -1
! 114: ELSE IF ( N .LT. 0 ) THEN
! 115: INFO = -2
! 116: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
! 117: INFO = -4
! 118: END IF
! 119: IF ( INFO .NE. 0 ) THEN
! 120: CALL XERBLA( 'DSYEQUB', -INFO )
! 121: RETURN
! 122: END IF
! 123:
! 124: UP = LSAME( UPLO, 'U' )
! 125: AMAX = ZERO
! 126: *
! 127: * Quick return if possible.
! 128: *
! 129: IF ( N .EQ. 0 ) THEN
! 130: SCOND = ONE
! 131: RETURN
! 132: END IF
! 133:
! 134: DO I = 1, N
! 135: S( I ) = ZERO
! 136: END DO
! 137:
! 138: AMAX = ZERO
! 139: IF ( UP ) THEN
! 140: DO J = 1, N
! 141: DO I = 1, J-1
! 142: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
! 143: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
! 144: AMAX = MAX( AMAX, ABS( A(I, J) ) )
! 145: END DO
! 146: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
! 147: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
! 148: END DO
! 149: ELSE
! 150: DO J = 1, N
! 151: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
! 152: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
! 153: DO I = J+1, N
! 154: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
! 155: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
! 156: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
! 157: END DO
! 158: END DO
! 159: END IF
! 160: DO J = 1, N
! 161: S( J ) = 1.0D+0 / S( J )
! 162: END DO
! 163:
! 164: TOL = ONE / SQRT(2.0D0 * N)
! 165:
! 166: DO ITER = 1, MAX_ITER
! 167: SCALE = 0.0D+0
! 168: SUMSQ = 0.0D+0
! 169: * BETA = |A|S
! 170: DO I = 1, N
! 171: WORK(I) = ZERO
! 172: END DO
! 173: IF ( UP ) THEN
! 174: DO J = 1, N
! 175: DO I = 1, J-1
! 176: T = ABS( A( I, J ) )
! 177: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
! 178: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
! 179: END DO
! 180: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
! 181: END DO
! 182: ELSE
! 183: DO J = 1, N
! 184: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
! 185: DO I = J+1, N
! 186: T = ABS( A( I, J ) )
! 187: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
! 188: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
! 189: END DO
! 190: END DO
! 191: END IF
! 192:
! 193: * avg = s^T beta / n
! 194: AVG = 0.0D+0
! 195: DO I = 1, N
! 196: AVG = AVG + S( I )*WORK( I )
! 197: END DO
! 198: AVG = AVG / N
! 199:
! 200: STD = 0.0D+0
! 201: DO I = 2*N+1, 3*N
! 202: WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
! 203: END DO
! 204: CALL DLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
! 205: STD = SCALE * SQRT( SUMSQ / N )
! 206:
! 207: IF ( STD .LT. TOL * AVG ) GOTO 999
! 208:
! 209: DO I = 1, N
! 210: T = ABS( A( I, I ) )
! 211: SI = S( I )
! 212: C2 = ( N-1 ) * T
! 213: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
! 214: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
! 215: D = C1*C1 - 4*C0*C2
! 216:
! 217: IF ( D .LE. 0 ) THEN
! 218: INFO = -1
! 219: RETURN
! 220: END IF
! 221: SI = -2*C0 / ( C1 + SQRT( D ) )
! 222:
! 223: D = SI - S( I )
! 224: U = ZERO
! 225: IF ( UP ) THEN
! 226: DO J = 1, I
! 227: T = ABS( A( J, I ) )
! 228: U = U + S( J )*T
! 229: WORK( J ) = WORK( J ) + D*T
! 230: END DO
! 231: DO J = I+1,N
! 232: T = ABS( A( I, J ) )
! 233: U = U + S( J )*T
! 234: WORK( J ) = WORK( J ) + D*T
! 235: END DO
! 236: ELSE
! 237: DO J = 1, I
! 238: T = ABS( A( I, J ) )
! 239: U = U + S( J )*T
! 240: WORK( J ) = WORK( J ) + D*T
! 241: END DO
! 242: DO J = I+1,N
! 243: T = ABS( A( J, I ) )
! 244: U = U + S( J )*T
! 245: WORK( J ) = WORK( J ) + D*T
! 246: END DO
! 247: END IF
! 248:
! 249: AVG = AVG + ( U + WORK( I ) ) * D / N
! 250: S( I ) = SI
! 251:
! 252: END DO
! 253:
! 254: END DO
! 255:
! 256: 999 CONTINUE
! 257:
! 258: SMLNUM = DLAMCH( 'SAFEMIN' )
! 259: BIGNUM = ONE / SMLNUM
! 260: SMIN = BIGNUM
! 261: SMAX = ZERO
! 262: T = ONE / SQRT(AVG)
! 263: BASE = DLAMCH( 'B' )
! 264: U = ONE / LOG( BASE )
! 265: DO I = 1, N
! 266: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
! 267: SMIN = MIN( SMIN, S( I ) )
! 268: SMAX = MAX( SMAX, S( I ) )
! 269: END DO
! 270: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 271: *
! 272: END
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