Annotation of rpl/lapack/lapack/zgbequb.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGBEQUB( 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 C( * ), R( * )
! 20: COMPLEX*16 AB( LDAB, * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * ZGBEQUB 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: * KL (input) INTEGER
! 54: * The number of subdiagonals within the band of A. KL >= 0.
! 55: *
! 56: * KU (input) INTEGER
! 57: * The number of superdiagonals within the band of A. KU >= 0.
! 58: *
! 59: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
! 60: * On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
! 61: * The j-th column of A is stored in the j-th column of the
! 62: * array AB as follows:
! 63: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
! 64: *
! 65: * LDAB (input) INTEGER
! 66: * The leading dimension of the array A. LDAB >= max(1,M).
! 67: *
! 68: * R (output) DOUBLE PRECISION array, dimension (M)
! 69: * If INFO = 0 or INFO > M, R contains the row scale factors
! 70: * for A.
! 71: *
! 72: * C (output) DOUBLE PRECISION array, dimension (N)
! 73: * If INFO = 0, C contains the column scale factors for A.
! 74: *
! 75: * ROWCND (output) DOUBLE PRECISION
! 76: * If INFO = 0 or INFO > M, ROWCND contains the ratio of the
! 77: * smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
! 78: * AMAX is neither too large nor too small, it is not worth
! 79: * scaling by R.
! 80: *
! 81: * COLCND (output) DOUBLE PRECISION
! 82: * If INFO = 0, COLCND contains the ratio of the smallest
! 83: * C(i) to the largest C(i). If COLCND >= 0.1, it is not
! 84: * worth scaling by C.
! 85: *
! 86: * AMAX (output) DOUBLE PRECISION
! 87: * Absolute value of largest matrix element. If AMAX is very
! 88: * close to overflow or very close to underflow, the matrix
! 89: * should be scaled.
! 90: *
! 91: * INFO (output) INTEGER
! 92: * = 0: successful exit
! 93: * < 0: if INFO = -i, the i-th argument had an illegal value
! 94: * > 0: if INFO = i, and i is
! 95: * <= M: the i-th row of A is exactly zero
! 96: * > M: the (i-M)-th column of A is exactly zero
! 97: *
! 98: * =====================================================================
! 99: *
! 100: * .. Parameters ..
! 101: DOUBLE PRECISION ONE, ZERO
! 102: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 103: * ..
! 104: * .. Local Scalars ..
! 105: INTEGER I, J, KD
! 106: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX,
! 107: $ LOGRDX
! 108: COMPLEX*16 ZDUM
! 109: * ..
! 110: * .. External Functions ..
! 111: DOUBLE PRECISION DLAMCH
! 112: EXTERNAL DLAMCH
! 113: * ..
! 114: * .. External Subroutines ..
! 115: EXTERNAL XERBLA
! 116: * ..
! 117: * .. Intrinsic Functions ..
! 118: INTRINSIC ABS, MAX, MIN, LOG, REAL, DIMAG
! 119: * ..
! 120: * .. Statement Functions ..
! 121: DOUBLE PRECISION CABS1
! 122: * ..
! 123: * .. Statement Function definitions ..
! 124: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 125: * ..
! 126: * .. Executable Statements ..
! 127: *
! 128: * Test the input parameters.
! 129: *
! 130: INFO = 0
! 131: IF( M.LT.0 ) THEN
! 132: INFO = -1
! 133: ELSE IF( N.LT.0 ) THEN
! 134: INFO = -2
! 135: ELSE IF( KL.LT.0 ) THEN
! 136: INFO = -3
! 137: ELSE IF( KU.LT.0 ) THEN
! 138: INFO = -4
! 139: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
! 140: INFO = -6
! 141: END IF
! 142: IF( INFO.NE.0 ) THEN
! 143: CALL XERBLA( 'ZGBEQUB', -INFO )
! 144: RETURN
! 145: END IF
! 146: *
! 147: * Quick return if possible.
! 148: *
! 149: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
! 150: ROWCND = ONE
! 151: COLCND = ONE
! 152: AMAX = ZERO
! 153: RETURN
! 154: END IF
! 155: *
! 156: * Get machine constants. Assume SMLNUM is a power of the radix.
! 157: *
! 158: SMLNUM = DLAMCH( 'S' )
! 159: BIGNUM = ONE / SMLNUM
! 160: RADIX = DLAMCH( 'B' )
! 161: LOGRDX = LOG(RADIX)
! 162: *
! 163: * Compute row scale factors.
! 164: *
! 165: DO 10 I = 1, M
! 166: R( I ) = ZERO
! 167: 10 CONTINUE
! 168: *
! 169: * Find the maximum element in each row.
! 170: *
! 171: KD = KU + 1
! 172: DO 30 J = 1, N
! 173: DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
! 174: R( I ) = MAX( R( I ), CABS1( AB( KD+I-J, J ) ) )
! 175: 20 CONTINUE
! 176: 30 CONTINUE
! 177: DO I = 1, M
! 178: IF( R( I ).GT.ZERO ) THEN
! 179: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
! 180: END IF
! 181: END DO
! 182: *
! 183: * Find the maximum and minimum scale factors.
! 184: *
! 185: RCMIN = BIGNUM
! 186: RCMAX = ZERO
! 187: DO 40 I = 1, M
! 188: RCMAX = MAX( RCMAX, R( I ) )
! 189: RCMIN = MIN( RCMIN, R( I ) )
! 190: 40 CONTINUE
! 191: AMAX = RCMAX
! 192: *
! 193: IF( RCMIN.EQ.ZERO ) THEN
! 194: *
! 195: * Find the first zero scale factor and return an error code.
! 196: *
! 197: DO 50 I = 1, M
! 198: IF( R( I ).EQ.ZERO ) THEN
! 199: INFO = I
! 200: RETURN
! 201: END IF
! 202: 50 CONTINUE
! 203: ELSE
! 204: *
! 205: * Invert the scale factors.
! 206: *
! 207: DO 60 I = 1, M
! 208: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
! 209: 60 CONTINUE
! 210: *
! 211: * Compute ROWCND = min(R(I)) / max(R(I)).
! 212: *
! 213: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 214: END IF
! 215: *
! 216: * Compute column scale factors.
! 217: *
! 218: DO 70 J = 1, N
! 219: C( J ) = ZERO
! 220: 70 CONTINUE
! 221: *
! 222: * Find the maximum element in each column,
! 223: * assuming the row scaling computed above.
! 224: *
! 225: DO 90 J = 1, N
! 226: DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
! 227: C( J ) = MAX( C( J ), CABS1( AB( KD+I-J, J ) )*R( I ) )
! 228: 80 CONTINUE
! 229: IF( C( J ).GT.ZERO ) THEN
! 230: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
! 231: END IF
! 232: 90 CONTINUE
! 233: *
! 234: * Find the maximum and minimum scale factors.
! 235: *
! 236: RCMIN = BIGNUM
! 237: RCMAX = ZERO
! 238: DO 100 J = 1, N
! 239: RCMIN = MIN( RCMIN, C( J ) )
! 240: RCMAX = MAX( RCMAX, C( J ) )
! 241: 100 CONTINUE
! 242: *
! 243: IF( RCMIN.EQ.ZERO ) THEN
! 244: *
! 245: * Find the first zero scale factor and return an error code.
! 246: *
! 247: DO 110 J = 1, N
! 248: IF( C( J ).EQ.ZERO ) THEN
! 249: INFO = M + J
! 250: RETURN
! 251: END IF
! 252: 110 CONTINUE
! 253: ELSE
! 254: *
! 255: * Invert the scale factors.
! 256: *
! 257: DO 120 J = 1, N
! 258: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
! 259: 120 CONTINUE
! 260: *
! 261: * Compute COLCND = min(C(J)) / max(C(J)).
! 262: *
! 263: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
! 264: END IF
! 265: *
! 266: RETURN
! 267: *
! 268: * End of ZGBEQUB
! 269: *
! 270: END
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