## MB01PD

### Matrix scaling (higher level routine)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To scale a matrix or undo scaling.  Scaling is performed, if
      necessary, so that the matrix norm will be in a safe range of
      representable numbers.

Specification

          SUBROUTINE MB01PD( SCUN, TYPE, M, N, KL, KU, ANRM, NBL, NROWS, A,
         $                   LDA, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          SCUN, TYPE
          INTEGER            INFO, KL, KU, LDA, M, MN, N, NBL
          DOUBLE PRECISION   ANRM
    C     .. Array Arguments ..
          INTEGER            NROWS ( * )
          DOUBLE PRECISION   A( LDA, * )

Arguments

Mode Parameters

      SCUN    CHARACTER*1
              SCUN indicates the operation to be performed.
              = 'S':  scale the matrix.
              = 'U':  undo scaling of the matrix.

      TYPE    CHARACTER*1
              TYPE indicates the storage type of the input matrix.
              = 'G':  A is a full matrix.
              = 'L':  A is a (block) lower triangular matrix.
              = 'U':  A is an (block) upper triangular matrix.
              = 'H':  A is an (block) upper Hessenberg matrix.
              = 'B':  A is a symmetric band matrix with lower bandwidth
                      KL and upper bandwidth KU and with the only the
                      lower half stored.
              = 'Q':  A is a symmetric band matrix with lower bandwidth
                      KL and upper bandwidth KU and with the only the
                      upper half stored.
              = 'Z':  A is a band matrix with lower bandwidth KL and
                      upper bandwidth KU.

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrix A. M >= 0.

      N       (input) INTEGER
              The number of columns of the matrix A. N >= 0.

      KL      (input) INTEGER
              The lower bandwidth of A.  Referenced only if TYPE = 'B',
              'Q' or 'Z'.

      KU      (input) INTEGER
              The upper bandwidth of A.  Referenced only if TYPE = 'B',
              'Q' or 'Z'.

      ANRM    (input) DOUBLE PRECISION
              The norm of the initial matrix A.  ANRM >= 0.
              When  ANRM = 0  then an immediate return is effected.
              ANRM should be preserved between the call of the routine
              with SCUN = 'S' and the corresponding one with SCUN = 'U'.

      NBL     (input) INTEGER
              The number of diagonal blocks of the matrix A, if it has a
              block structure.  To specify that matrix A has no block
              structure, set NBL = 0.  NBL >= 0.

      NROWS   (input) INTEGER array, dimension max(1,NBL)
              NROWS(i) contains the number of rows and columns of the
              i-th diagonal block of matrix A.  The sum of the values
              NROWS(i),  for  i = 1: NBL,  should be equal to min(M,N).
              The elements of the array  NROWS  are not referenced if
              NBL = 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M by N part of this array must
              contain the matrix to be scaled/unscaled.
              On exit, the leading M by N part of A will contain
              the modified matrix.
              The storage mode of A is specified by TYPE.

      LDA     (input) INTEGER
              The leading dimension of the array A.  LDA  >= max(1,M).

Error Indicator

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      Denote by ANRM the norm of the matrix, and by SMLNUM and BIGNUM,
      two positive numbers near the smallest and largest safely
      representable numbers, respectively.  The matrix is scaled, if
      needed, such that the norm of the result is in the range
      [SMLNUM, BIGNUM].  The scaling factor is represented as a ratio
      of two numbers, one of them being ANRM, and the other one either
      SMLNUM or BIGNUM, depending on ANRM being less than SMLNUM or
      larger than BIGNUM, respectively.  For undoing the scaling, the
      norm is again compared with SMLNUM or BIGNUM, and the reciprocal
      of the previous scaling factor is used.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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