LAPACK 3.3.1 Linear Algebra PACKage

# dgbequb.f

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```00001       SUBROUTINE DGBEQUB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
00002      \$                    AMAX, INFO )
00003 *
00004 *     -- LAPACK routine (version 3.2)                                 --
00005 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
00006 *     -- Jason Riedy of Univ. of California Berkeley.                 --
00007 *     -- November 2008                                                --
00008 *
00009 *     -- LAPACK is a software package provided by Univ. of Tennessee, --
00010 *     -- Univ. of California Berkeley and NAG Ltd.                    --
00011 *
00012       IMPLICIT NONE
00013 *     ..
00014 *     .. Scalar Arguments ..
00015       INTEGER            INFO, KL, KU, LDAB, M, N
00016       DOUBLE PRECISION   AMAX, COLCND, ROWCND
00017 *     ..
00018 *     .. Array Arguments ..
00019       DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  DGBEQUB computes row and column scalings intended to equilibrate an
00026 *  M-by-N matrix A and reduce its condition number.  R returns the row
00027 *  scale factors and C the column scale factors, chosen to try to make
00028 *  the largest element in each row and column of the matrix B with
00029 *  elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
00031 *
00032 *  R(i) and C(j) are restricted to be a power of the radix between
00033 *  SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
00034 *  of these scaling factors is not guaranteed to reduce the condition
00035 *  number of A but works well in practice.
00036 *
00037 *  This routine differs from DGEEQU by restricting the scaling factors
00038 *  to a power of the radix.  Baring over- and underflow, scaling by
00039 *  these factors introduces no additional rounding errors.  However, the
00040 *  scaled entries' magnitured are no longer approximately 1 but lie
00042 *
00043 *  Arguments
00044 *  =========
00045 *
00046 *  M       (input) INTEGER
00047 *          The number of rows of the matrix A.  M >= 0.
00048 *
00049 *  N       (input) INTEGER
00050 *          The number of columns of the matrix A.  N >= 0.
00051 *
00052 *  KL      (input) INTEGER
00053 *          The number of subdiagonals within the band of A.  KL >= 0.
00054 *
00055 *  KU      (input) INTEGER
00056 *          The number of superdiagonals within the band of A.  KU >= 0.
00057 *
00058 *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
00059 *          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
00060 *          The j-th column of A is stored in the j-th column of the
00061 *          array AB as follows:
00062 *          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
00063 *
00064 *  LDAB    (input) INTEGER
00065 *          The leading dimension of the array A.  LDAB >= max(1,M).
00066 *
00067 *  R       (output) DOUBLE PRECISION array, dimension (M)
00068 *          If INFO = 0 or INFO > M, R contains the row scale factors
00069 *          for A.
00070 *
00071 *  C       (output) DOUBLE PRECISION array, dimension (N)
00072 *          If INFO = 0,  C contains the column scale factors for A.
00073 *
00074 *  ROWCND  (output) DOUBLE PRECISION
00075 *          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
00076 *          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
00077 *          AMAX is neither too large nor too small, it is not worth
00078 *          scaling by R.
00079 *
00080 *  COLCND  (output) DOUBLE PRECISION
00081 *          If INFO = 0, COLCND contains the ratio of the smallest
00082 *          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
00083 *          worth scaling by C.
00084 *
00085 *  AMAX    (output) DOUBLE PRECISION
00086 *          Absolute value of largest matrix element.  If AMAX is very
00087 *          close to overflow or very close to underflow, the matrix
00088 *          should be scaled.
00089 *
00090 *  INFO    (output) INTEGER
00091 *          = 0:  successful exit
00092 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00093 *          > 0:  if INFO = i,  and i is
00094 *                <= M:  the i-th row of A is exactly zero
00095 *                >  M:  the (i-M)-th column of A is exactly zero
00096 *
00097 *  =====================================================================
00098 *
00099 *     .. Parameters ..
00100       DOUBLE PRECISION   ONE, ZERO
00101       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00102 *     ..
00103 *     .. Local Scalars ..
00104       INTEGER            I, J, KD
00105       DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
00106 *     ..
00107 *     .. External Functions ..
00108       DOUBLE PRECISION   DLAMCH
00109       EXTERNAL           DLAMCH
00110 *     ..
00111 *     .. External Subroutines ..
00112       EXTERNAL           XERBLA
00113 *     ..
00114 *     .. Intrinsic Functions ..
00115       INTRINSIC          ABS, MAX, MIN, LOG
00116 *     ..
00117 *     .. Executable Statements ..
00118 *
00119 *     Test the input parameters.
00120 *
00121       INFO = 0
00122       IF( M.LT.0 ) THEN
00123          INFO = -1
00124       ELSE IF( N.LT.0 ) THEN
00125          INFO = -2
00126       ELSE IF( KL.LT.0 ) THEN
00127          INFO = -3
00128       ELSE IF( KU.LT.0 ) THEN
00129          INFO = -4
00130       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
00131          INFO = -6
00132       END IF
00133       IF( INFO.NE.0 ) THEN
00134          CALL XERBLA( 'DGBEQUB', -INFO )
00135          RETURN
00136       END IF
00137 *
00138 *     Quick return if possible.
00139 *
00140       IF( M.EQ.0 .OR. N.EQ.0 ) THEN
00141          ROWCND = ONE
00142          COLCND = ONE
00143          AMAX = ZERO
00144          RETURN
00145       END IF
00146 *
00147 *     Get machine constants.  Assume SMLNUM is a power of the radix.
00148 *
00149       SMLNUM = DLAMCH( 'S' )
00150       BIGNUM = ONE / SMLNUM
00151       RADIX = DLAMCH( 'B' )
00153 *
00154 *     Compute row scale factors.
00155 *
00156       DO 10 I = 1, M
00157          R( I ) = ZERO
00158    10 CONTINUE
00159 *
00160 *     Find the maximum element in each row.
00161 *
00162       KD = KU + 1
00163       DO 30 J = 1, N
00164          DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
00165             R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
00166    20    CONTINUE
00167    30 CONTINUE
00168       DO I = 1, M
00169          IF( R( I ).GT.ZERO ) THEN
00170             R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
00171          END IF
00172       END DO
00173 *
00174 *     Find the maximum and minimum scale factors.
00175 *
00176       RCMIN = BIGNUM
00177       RCMAX = ZERO
00178       DO 40 I = 1, M
00179          RCMAX = MAX( RCMAX, R( I ) )
00180          RCMIN = MIN( RCMIN, R( I ) )
00181    40 CONTINUE
00182       AMAX = RCMAX
00183 *
00184       IF( RCMIN.EQ.ZERO ) THEN
00185 *
00186 *        Find the first zero scale factor and return an error code.
00187 *
00188          DO 50 I = 1, M
00189             IF( R( I ).EQ.ZERO ) THEN
00190                INFO = I
00191                RETURN
00192             END IF
00193    50    CONTINUE
00194       ELSE
00195 *
00196 *        Invert the scale factors.
00197 *
00198          DO 60 I = 1, M
00199             R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
00200    60    CONTINUE
00201 *
00202 *        Compute ROWCND = min(R(I)) / max(R(I)).
00203 *
00204          ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00205       END IF
00206 *
00207 *     Compute column scale factors.
00208 *
00209       DO 70 J = 1, N
00210          C( J ) = ZERO
00211    70 CONTINUE
00212 *
00213 *     Find the maximum element in each column,
00214 *     assuming the row scaling computed above.
00215 *
00216       DO 90 J = 1, N
00217          DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
00218             C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
00219    80    CONTINUE
00220          IF( C( J ).GT.ZERO ) THEN
00221             C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
00222          END IF
00223    90 CONTINUE
00224 *
00225 *     Find the maximum and minimum scale factors.
00226 *
00227       RCMIN = BIGNUM
00228       RCMAX = ZERO
00229       DO 100 J = 1, N
00230          RCMIN = MIN( RCMIN, C( J ) )
00231          RCMAX = MAX( RCMAX, C( J ) )
00232   100 CONTINUE
00233 *
00234       IF( RCMIN.EQ.ZERO ) THEN
00235 *
00236 *        Find the first zero scale factor and return an error code.
00237 *
00238          DO 110 J = 1, N
00239             IF( C( J ).EQ.ZERO ) THEN
00240                INFO = M + J
00241                RETURN
00242             END IF
00243   110    CONTINUE
00244       ELSE
00245 *
00246 *        Invert the scale factors.
00247 *
00248          DO 120 J = 1, N
00249             C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
00250   120    CONTINUE
00251 *
00252 *        Compute COLCND = min(C(J)) / max(C(J)).
00253 *
00254          COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
00255       END IF
00256 *
00257       RETURN
00258 *
00259 *     End of DGBEQUB
00260 *
00261       END
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