Annotation of rpl/lapack/lapack/zgeequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
! 2: $ INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
! 6: * -- Jason Riedy of Univ. of California Berkeley. --
! 7: * -- November 2008 --
! 8: *
! 9: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 10: * -- Univ. of California Berkeley and NAG Ltd. --
! 11: *
! 12: IMPLICIT NONE
! 13: * ..
! 14: * .. Scalar Arguments ..
! 15: INTEGER INFO, LDA, M, N
! 16: DOUBLE PRECISION AMAX, COLCND, ROWCND
! 17: * ..
! 18: * .. Array Arguments ..
! 19: DOUBLE PRECISION C( * ), R( * )
! 20: COMPLEX*16 A( LDA, * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * ZGEEQUB computes row and column scalings intended to equilibrate an
! 27: * M-by-N matrix A and reduce its condition number. R returns the row
! 28: * scale factors and C the column scale factors, chosen to try to make
! 29: * the largest element in each row and column of the matrix B with
! 30: * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
! 31: * the radix.
! 32: *
! 33: * R(i) and C(j) are restricted to be a power of the radix between
! 34: * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
! 35: * of these scaling factors is not guaranteed to reduce the condition
! 36: * number of A but works well in practice.
! 37: *
! 38: * This routine differs from ZGEEQU by restricting the scaling factors
! 39: * to a power of the radix. Baring over- and underflow, scaling by
! 40: * these factors introduces no additional rounding errors. However, the
! 41: * scaled entries' magnitured are no longer approximately 1 but lie
! 42: * between sqrt(radix) and 1/sqrt(radix).
! 43: *
! 44: * Arguments
! 45: * =========
! 46: *
! 47: * M (input) INTEGER
! 48: * The number of rows of the matrix A. M >= 0.
! 49: *
! 50: * N (input) INTEGER
! 51: * The number of columns of the matrix A. N >= 0.
! 52: *
! 53: * A (input) COMPLEX*16 array, dimension (LDA,N)
! 54: * The M-by-N matrix whose equilibration factors are
! 55: * to be computed.
! 56: *
! 57: * LDA (input) INTEGER
! 58: * The leading dimension of the array A. LDA >= max(1,M).
! 59: *
! 60: * R (output) DOUBLE PRECISION array, dimension (M)
! 61: * If INFO = 0 or INFO > M, R contains the row scale factors
! 62: * for A.
! 63: *
! 64: * C (output) DOUBLE PRECISION array, dimension (N)
! 65: * If INFO = 0, C contains the column scale factors for A.
! 66: *
! 67: * ROWCND (output) DOUBLE PRECISION
! 68: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 69: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 70: * AMAX is neither too large nor too small, it is not worth
! 71: * scaling by R.
! 72: *
! 73: * COLCND (output) DOUBLE PRECISION
! 74: * If INFO = 0, COLCND contains the ratio of the smallest
! 75: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 76: * worth scaling by C.
! 77: *
! 78: * AMAX (output) DOUBLE PRECISION
! 79: * Absolute value of largest matrix element. If AMAX is very
! 80: * close to overflow or very close to underflow, the matrix
! 81: * should be scaled.
! 82: *
! 83: * INFO (output) INTEGER
! 84: * = 0: successful exit
! 85: * < 0: if INFO = -i, the i-th argument had an illegal value
! 86: * > 0: if INFO = i, and i is
! 87: * <= M: the i-th row of A is exactly zero
! 88: * > M: the (i-M)-th column of A is exactly zero
! 89: *
! 90: * =====================================================================
! 91: *
! 92: * .. Parameters ..
! 93: DOUBLE PRECISION ONE, ZERO
! 94: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 95: * ..
! 96: * .. Local Scalars ..
! 97: INTEGER I, J
! 98: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
! 99: COMPLEX*16 ZDUM
! 100: * ..
! 101: * .. External Functions ..
! 102: DOUBLE PRECISION DLAMCH
! 103: EXTERNAL DLAMCH
! 104: * ..
! 105: * .. External Subroutines ..
! 106: EXTERNAL XERBLA
! 107: * ..
! 108: * .. Intrinsic Functions ..
! 109: INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
! 110: * ..
! 111: * .. Statement Functions ..
! 112: DOUBLE PRECISION CABS1
! 113: * ..
! 114: * .. Statement Function definitions ..
! 115: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 116: * ..
! 117: * .. Executable Statements ..
! 118: *
! 119: * Test the input parameters.
! 120: *
! 121: INFO = 0
! 122: IF( M.LT.0 ) THEN
! 123: INFO = -1
! 124: ELSE IF( N.LT.0 ) THEN
! 125: INFO = -2
! 126: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 127: INFO = -4
! 128: END IF
! 129: IF( INFO.NE.0 ) THEN
! 130: CALL XERBLA( 'ZGEEQUB', -INFO )
! 131: RETURN
! 132: END IF
! 133: *
! 134: * Quick return if possible.
! 135: *
! 136: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
! 137: ROWCND = ONE
! 138: COLCND = ONE
! 139: AMAX = ZERO
! 140: RETURN
! 141: END IF
! 142: *
! 143: * Get machine constants. Assume SMLNUM is a power of the radix.
! 144: *
! 145: SMLNUM = DLAMCH( 'S' )
! 146: BIGNUM = ONE / SMLNUM
! 147: RADIX = DLAMCH( 'B' )
! 148: LOGRDX = LOG( RADIX )
! 149: *
! 150: * Compute row scale factors.
! 151: *
! 152: DO 10 I = 1, M
! 153: R( I ) = ZERO
! 154: 10 CONTINUE
! 155: *
! 156: * Find the maximum element in each row.
! 157: *
! 158: DO 30 J = 1, N
! 159: DO 20 I = 1, M
! 160: R( I ) = MAX( R( I ), CABS1( A( I, J ) ) )
! 161: 20 CONTINUE
! 162: 30 CONTINUE
! 163: DO I = 1, M
! 164: IF( R( I ).GT.ZERO ) THEN
! 165: R( I ) = RADIX**INT( LOG(R( I ) ) / LOGRDX )
! 166: END IF
! 167: END DO
! 168: *
! 169: * Find the maximum and minimum scale factors.
! 170: *
! 171: RCMIN = BIGNUM
! 172: RCMAX = ZERO
! 173: DO 40 I = 1, M
! 174: RCMAX = MAX( RCMAX, R( I ) )
! 175: RCMIN = MIN( RCMIN, R( I ) )
! 176: 40 CONTINUE
! 177: AMAX = RCMAX
! 178: *
! 179: IF( RCMIN.EQ.ZERO ) THEN
! 180: *
! 181: * Find the first zero scale factor and return an error code.
! 182: *
! 183: DO 50 I = 1, M
! 184: IF( R( I ).EQ.ZERO ) THEN
! 185: INFO = I
! 186: RETURN
! 187: END IF
! 188: 50 CONTINUE
! 189: ELSE
! 190: *
! 191: * Invert the scale factors.
! 192: *
! 193: DO 60 I = 1, M
! 194: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
! 195: 60 CONTINUE
! 196: *
! 197: * Compute ROWCND = min(R(I)) / max(R(I)).
! 198: *
! 199: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 200: END IF
! 201: *
! 202: * Compute column scale factors.
! 203: *
! 204: DO 70 J = 1, N
! 205: C( J ) = ZERO
! 206: 70 CONTINUE
! 207: *
! 208: * Find the maximum element in each column,
! 209: * assuming the row scaling computed above.
! 210: *
! 211: DO 90 J = 1, N
! 212: DO 80 I = 1, M
! 213: C( J ) = MAX( C( J ), CABS1( A( I, J ) )*R( I ) )
! 214: 80 CONTINUE
! 215: IF( C( J ).GT.ZERO ) THEN
! 216: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
! 217: END IF
! 218: 90 CONTINUE
! 219: *
! 220: * Find the maximum and minimum scale factors.
! 221: *
! 222: RCMIN = BIGNUM
! 223: RCMAX = ZERO
! 224: DO 100 J = 1, N
! 225: RCMIN = MIN( RCMIN, C( J ) )
! 226: RCMAX = MAX( RCMAX, C( J ) )
! 227: 100 CONTINUE
! 228: *
! 229: IF( RCMIN.EQ.ZERO ) THEN
! 230: *
! 231: * Find the first zero scale factor and return an error code.
! 232: *
! 233: DO 110 J = 1, N
! 234: IF( C( J ).EQ.ZERO ) THEN
! 235: INFO = M + J
! 236: RETURN
! 237: END IF
! 238: 110 CONTINUE
! 239: ELSE
! 240: *
! 241: * Invert the scale factors.
! 242: *
! 243: DO 120 J = 1, N
! 244: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
! 245: 120 CONTINUE
! 246: *
! 247: * Compute COLCND = min(C(J)) / max(C(J)).
! 248: *
! 249: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 250: END IF
! 251: *
! 252: RETURN
! 253: *
! 254: * End of ZGEEQUB
! 255: *
! 256: END
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