Annotation of rpl/lapack/lapack/dgeequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGEEQUB( 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 A( LDA, * ), C( * ), R( * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DGEEQUB computes row and column scalings intended to equilibrate an
! 26: * M-by-N matrix A and reduce its condition number. R returns the row
! 27: * scale factors and C the column scale factors, chosen to try to make
! 28: * the largest element in each row and column of the matrix B with
! 29: * elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
! 30: * the radix.
! 31: *
! 32: * R(i) and C(j) are restricted to be a power of the radix between
! 33: * SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
! 34: * of these scaling factors is not guaranteed to reduce the condition
! 35: * number of A but works well in practice.
! 36: *
! 37: * This routine differs from DGEEQU by restricting the scaling factors
! 38: * to a power of the radix. Baring over- and underflow, scaling by
! 39: * these factors introduces no additional rounding errors. However, the
! 40: * scaled entries' magnitured are no longer approximately 1 but lie
! 41: * between sqrt(radix) and 1/sqrt(radix).
! 42: *
! 43: * Arguments
! 44: * =========
! 45: *
! 46: * M (input) INTEGER
! 47: * The number of rows of the matrix A. M >= 0.
! 48: *
! 49: * N (input) INTEGER
! 50: * The number of columns of the matrix A. N >= 0.
! 51: *
! 52: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 53: * The M-by-N matrix whose equilibration factors are
! 54: * to be computed.
! 55: *
! 56: * LDA (input) INTEGER
! 57: * The leading dimension of the array A. LDA >= max(1,M).
! 58: *
! 59: * R (output) DOUBLE PRECISION array, dimension (M)
! 60: * If INFO = 0 or INFO > M, R contains the row scale factors
! 61: * for A.
! 62: *
! 63: * C (output) DOUBLE PRECISION array, dimension (N)
! 64: * If INFO = 0, C contains the column scale factors for A.
! 65: *
! 66: * ROWCND (output) DOUBLE PRECISION
! 67: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 68: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 69: * AMAX is neither too large nor too small, it is not worth
! 70: * scaling by R.
! 71: *
! 72: * COLCND (output) DOUBLE PRECISION
! 73: * If INFO = 0, COLCND contains the ratio of the smallest
! 74: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 75: * worth scaling by C.
! 76: *
! 77: * AMAX (output) DOUBLE PRECISION
! 78: * Absolute value of largest matrix element. If AMAX is very
! 79: * close to overflow or very close to underflow, the matrix
! 80: * should be scaled.
! 81: *
! 82: * INFO (output) INTEGER
! 83: * = 0: successful exit
! 84: * < 0: if INFO = -i, the i-th argument had an illegal value
! 85: * > 0: if INFO = i, and i is
! 86: * <= M: the i-th row of A is exactly zero
! 87: * > M: the (i-M)-th column of A is exactly zero
! 88: *
! 89: * =====================================================================
! 90: *
! 91: * .. Parameters ..
! 92: DOUBLE PRECISION ONE, ZERO
! 93: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 94: * ..
! 95: * .. Local Scalars ..
! 96: INTEGER I, J
! 97: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
! 98: * ..
! 99: * .. External Functions ..
! 100: DOUBLE PRECISION DLAMCH
! 101: EXTERNAL DLAMCH
! 102: * ..
! 103: * .. External Subroutines ..
! 104: EXTERNAL XERBLA
! 105: * ..
! 106: * .. Intrinsic Functions ..
! 107: INTRINSIC ABS, MAX, MIN, LOG
! 108: * ..
! 109: * .. Executable Statements ..
! 110: *
! 111: * Test the input parameters.
! 112: *
! 113: INFO = 0
! 114: IF( M.LT.0 ) THEN
! 115: INFO = -1
! 116: ELSE IF( N.LT.0 ) THEN
! 117: INFO = -2
! 118: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
! 119: INFO = -4
! 120: END IF
! 121: IF( INFO.NE.0 ) THEN
! 122: CALL XERBLA( 'DGEEQUB', -INFO )
! 123: RETURN
! 124: END IF
! 125: *
! 126: * Quick return if possible.
! 127: *
! 128: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
! 129: ROWCND = ONE
! 130: COLCND = ONE
! 131: AMAX = ZERO
! 132: RETURN
! 133: END IF
! 134: *
! 135: * Get machine constants. Assume SMLNUM is a power of the radix.
! 136: *
! 137: SMLNUM = DLAMCH( 'S' )
! 138: BIGNUM = ONE / SMLNUM
! 139: RADIX = DLAMCH( 'B' )
! 140: LOGRDX = LOG( RADIX )
! 141: *
! 142: * Compute row scale factors.
! 143: *
! 144: DO 10 I = 1, M
! 145: R( I ) = ZERO
! 146: 10 CONTINUE
! 147: *
! 148: * Find the maximum element in each row.
! 149: *
! 150: DO 30 J = 1, N
! 151: DO 20 I = 1, M
! 152: R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
! 153: 20 CONTINUE
! 154: 30 CONTINUE
! 155: DO I = 1, M
! 156: IF( R( I ).GT.ZERO ) THEN
! 157: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
! 158: END IF
! 159: END DO
! 160: *
! 161: * Find the maximum and minimum scale factors.
! 162: *
! 163: RCMIN = BIGNUM
! 164: RCMAX = ZERO
! 165: DO 40 I = 1, M
! 166: RCMAX = MAX( RCMAX, R( I ) )
! 167: RCMIN = MIN( RCMIN, R( I ) )
! 168: 40 CONTINUE
! 169: AMAX = RCMAX
! 170: *
! 171: IF( RCMIN.EQ.ZERO ) THEN
! 172: *
! 173: * Find the first zero scale factor and return an error code.
! 174: *
! 175: DO 50 I = 1, M
! 176: IF( R( I ).EQ.ZERO ) THEN
! 177: INFO = I
! 178: RETURN
! 179: END IF
! 180: 50 CONTINUE
! 181: ELSE
! 182: *
! 183: * Invert the scale factors.
! 184: *
! 185: DO 60 I = 1, M
! 186: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
! 187: 60 CONTINUE
! 188: *
! 189: * Compute ROWCND = min(R(I)) / max(R(I)).
! 190: *
! 191: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 192: END IF
! 193: *
! 194: * Compute column scale factors
! 195: *
! 196: DO 70 J = 1, N
! 197: C( J ) = ZERO
! 198: 70 CONTINUE
! 199: *
! 200: * Find the maximum element in each column,
! 201: * assuming the row scaling computed above.
! 202: *
! 203: DO 90 J = 1, N
! 204: DO 80 I = 1, M
! 205: C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
! 206: 80 CONTINUE
! 207: IF( C( J ).GT.ZERO ) THEN
! 208: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
! 209: END IF
! 210: 90 CONTINUE
! 211: *
! 212: * Find the maximum and minimum scale factors.
! 213: *
! 214: RCMIN = BIGNUM
! 215: RCMAX = ZERO
! 216: DO 100 J = 1, N
! 217: RCMIN = MIN( RCMIN, C( J ) )
! 218: RCMAX = MAX( RCMAX, C( J ) )
! 219: 100 CONTINUE
! 220: *
! 221: IF( RCMIN.EQ.ZERO ) THEN
! 222: *
! 223: * Find the first zero scale factor and return an error code.
! 224: *
! 225: DO 110 J = 1, N
! 226: IF( C( J ).EQ.ZERO ) THEN
! 227: INFO = M + J
! 228: RETURN
! 229: END IF
! 230: 110 CONTINUE
! 231: ELSE
! 232: *
! 233: * Invert the scale factors.
! 234: *
! 235: DO 120 J = 1, N
! 236: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
! 237: 120 CONTINUE
! 238: *
! 239: * Compute COLCND = min(C(J)) / max(C(J)).
! 240: *
! 241: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 242: END IF
! 243: *
! 244: RETURN
! 245: *
! 246: * End of DGEEQUB
! 247: *
! 248: END
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