Annotation of rpl/lapack/lapack/dgbequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
! 2: $ AMAX, 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, KL, KU, LDAB, M, N
! 16: DOUBLE PRECISION AMAX, COLCND, ROWCND
! 17: * ..
! 18: * .. Array Arguments ..
! 19: DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DGBEQUB 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: * KL (input) INTEGER
! 53: * The number of subdiagonals within the band of A. KL >= 0.
! 54: *
! 55: * KU (input) INTEGER
! 56: * The number of superdiagonals within the band of A. KU >= 0.
! 57: *
! 58: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
! 59: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
! 60: * The j-th column of A is stored in the j-th column of the
! 61: * array AB as follows:
! 62: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
! 63: *
! 64: * LDAB (input) INTEGER
! 65: * The leading dimension of the array A. LDAB >= max(1,M).
! 66: *
! 67: * R (output) DOUBLE PRECISION array, dimension (M)
! 68: * If INFO = 0 or INFO > M, R contains the row scale factors
! 69: * for A.
! 70: *
! 71: * C (output) DOUBLE PRECISION array, dimension (N)
! 72: * If INFO = 0, C contains the column scale factors for A.
! 73: *
! 74: * ROWCND (output) DOUBLE PRECISION
! 75: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 76: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 77: * AMAX is neither too large nor too small, it is not worth
! 78: * scaling by R.
! 79: *
! 80: * COLCND (output) DOUBLE PRECISION
! 81: * If INFO = 0, COLCND contains the ratio of the smallest
! 82: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 83: * worth scaling by C.
! 84: *
! 85: * AMAX (output) DOUBLE PRECISION
! 86: * Absolute value of largest matrix element. If AMAX is very
! 87: * close to overflow or very close to underflow, the matrix
! 88: * should be scaled.
! 89: *
! 90: * INFO (output) INTEGER
! 91: * = 0: successful exit
! 92: * < 0: if INFO = -i, the i-th argument had an illegal value
! 93: * > 0: if INFO = i, and i is
! 94: * <= M: the i-th row of A is exactly zero
! 95: * > M: the (i-M)-th column of A is exactly zero
! 96: *
! 97: * =====================================================================
! 98: *
! 99: * .. Parameters ..
! 100: DOUBLE PRECISION ONE, ZERO
! 101: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 102: * ..
! 103: * .. Local Scalars ..
! 104: INTEGER I, J, KD
! 105: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
! 106: * ..
! 107: * .. External Functions ..
! 108: DOUBLE PRECISION DLAMCH
! 109: EXTERNAL DLAMCH
! 110: * ..
! 111: * .. External Subroutines ..
! 112: EXTERNAL XERBLA
! 113: * ..
! 114: * .. Intrinsic Functions ..
! 115: INTRINSIC ABS, MAX, MIN, LOG
! 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( KL.LT.0 ) THEN
! 127: INFO = -3
! 128: ELSE IF( KU.LT.0 ) THEN
! 129: INFO = -4
! 130: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
! 131: INFO = -6
! 132: END IF
! 133: IF( INFO.NE.0 ) THEN
! 134: CALL XERBLA( 'DGBEQUB', -INFO )
! 135: RETURN
! 136: END IF
! 137: *
! 138: * Quick return if possible.
! 139: *
! 140: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
! 141: ROWCND = ONE
! 142: COLCND = ONE
! 143: AMAX = ZERO
! 144: RETURN
! 145: END IF
! 146: *
! 147: * Get machine constants. Assume SMLNUM is a power of the radix.
! 148: *
! 149: SMLNUM = DLAMCH( 'S' )
! 150: BIGNUM = ONE / SMLNUM
! 151: RADIX = DLAMCH( 'B' )
! 152: LOGRDX = LOG(RADIX)
! 153: *
! 154: * Compute row scale factors.
! 155: *
! 156: DO 10 I = 1, M
! 157: R( I ) = ZERO
! 158: 10 CONTINUE
! 159: *
! 160: * Find the maximum element in each row.
! 161: *
! 162: KD = KU + 1
! 163: DO 30 J = 1, N
! 164: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
! 165: R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
! 166: 20 CONTINUE
! 167: 30 CONTINUE
! 168: DO I = 1, M
! 169: IF( R( I ).GT.ZERO ) THEN
! 170: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
! 171: END IF
! 172: END DO
! 173: *
! 174: * Find the maximum and minimum scale factors.
! 175: *
! 176: RCMIN = BIGNUM
! 177: RCMAX = ZERO
! 178: DO 40 I = 1, M
! 179: RCMAX = MAX( RCMAX, R( I ) )
! 180: RCMIN = MIN( RCMIN, R( I ) )
! 181: 40 CONTINUE
! 182: AMAX = RCMAX
! 183: *
! 184: IF( RCMIN.EQ.ZERO ) THEN
! 185: *
! 186: * Find the first zero scale factor and return an error code.
! 187: *
! 188: DO 50 I = 1, M
! 189: IF( R( I ).EQ.ZERO ) THEN
! 190: INFO = I
! 191: RETURN
! 192: END IF
! 193: 50 CONTINUE
! 194: ELSE
! 195: *
! 196: * Invert the scale factors.
! 197: *
! 198: DO 60 I = 1, M
! 199: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
! 200: 60 CONTINUE
! 201: *
! 202: * Compute ROWCND = min(R(I)) / max(R(I)).
! 203: *
! 204: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 205: END IF
! 206: *
! 207: * Compute column scale factors.
! 208: *
! 209: DO 70 J = 1, N
! 210: C( J ) = ZERO
! 211: 70 CONTINUE
! 212: *
! 213: * Find the maximum element in each column,
! 214: * assuming the row scaling computed above.
! 215: *
! 216: DO 90 J = 1, N
! 217: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
! 218: C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
! 219: 80 CONTINUE
! 220: IF( C( J ).GT.ZERO ) THEN
! 221: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
! 222: END IF
! 223: 90 CONTINUE
! 224: *
! 225: * Find the maximum and minimum scale factors.
! 226: *
! 227: RCMIN = BIGNUM
! 228: RCMAX = ZERO
! 229: DO 100 J = 1, N
! 230: RCMIN = MIN( RCMIN, C( J ) )
! 231: RCMAX = MAX( RCMAX, C( J ) )
! 232: 100 CONTINUE
! 233: *
! 234: IF( RCMIN.EQ.ZERO ) THEN
! 235: *
! 236: * Find the first zero scale factor and return an error code.
! 237: *
! 238: DO 110 J = 1, N
! 239: IF( C( J ).EQ.ZERO ) THEN
! 240: INFO = M + J
! 241: RETURN
! 242: END IF
! 243: 110 CONTINUE
! 244: ELSE
! 245: *
! 246: * Invert the scale factors.
! 247: *
! 248: DO 120 J = 1, N
! 249: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
! 250: 120 CONTINUE
! 251: *
! 252: * Compute COLCND = min(C(J)) / max(C(J)).
! 253: *
! 254: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 255: END IF
! 256: *
! 257: RETURN
! 258: *
! 259: * End of DGBEQUB
! 260: *
! 261: END
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