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NAME

       realGBauxiliary

SYNOPSIS

   Functions
       real function slangb (NORM, N, KL, KU, AB, LDAB, WORK)
           SLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
           absolute value of any element of general band matrix.
       subroutine slaqgb (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
           SLAQGB scales a general band matrix, using row and column scaling factors computed by
           sgbequ.

Detailed Description

       This is the group of real auxiliary functions for GB matrices

Function Documentation

   real function slangb (character NORM, integer N, integer KL, integer KU, real, dimension(
       ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)
       SLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
       absolute value of any element of general band matrix.

       Purpose:

            SLANGB  returns the value of the one norm,  or the Frobenius norm, or
            the  infinity norm,  or the element of  largest absolute value  of an
            n by n band matrix  A,  with kl sub-diagonals and ku super-diagonals.

       Returns:
           SLANGB

               SLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                        (
                        ( norm1(A),         NORM = '1', 'O' or 'o'
                        (
                        ( normI(A),         NORM = 'I' or 'i'
                        (
                        ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

            where  norm1  denotes the  one norm of a matrix (maximum column sum),
            normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
            normF  denotes the  Frobenius norm of a matrix (square root of sum of
            squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

       Parameters:
           NORM

                     NORM is CHARACTER*1
                     Specifies the value to be returned in SLANGB as described
                     above.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.  When N = 0, SLANGB is
                     set to zero.

           KL

                     KL is INTEGER
                     The number of sub-diagonals of the matrix A.  KL >= 0.

           KU

                     KU is INTEGER
                     The number of super-diagonals of the matrix A.  KU >= 0.

           AB

                     AB is REAL array, dimension (LDAB,N)
                     The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
                     column of A is stored in the j-th column of the array AB as
                     follows:
                     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KL+KU+1.

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK)),
                     where LWORK >= N when NORM = 'I'; otherwise, WORK is not
                     referenced.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine slaqgb (integer M, integer N, integer KL, integer KU, real, dimension( ldab, * )
       AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real
       COLCND, real AMAX, character EQUED)
       SLAQGB scales a general band matrix, using row and column scaling factors computed by
       sgbequ.

       Purpose:

            SLAQGB equilibrates a general M by N band matrix A with KL
            subdiagonals and KU superdiagonals using the row and scaling factors
            in the vectors R and C.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           KL

                     KL is INTEGER
                     The number of subdiagonals within the band of A.  KL >= 0.

           KU

                     KU is INTEGER
                     The number of superdiagonals within the band of A.  KU >= 0.

           AB

                     AB is REAL array, dimension (LDAB,N)
                     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
                     The j-th column of A is stored in the j-th column of the
                     array AB as follows:
                     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

                     On exit, the equilibrated matrix, in the same storage format
                     as A.  See EQUED for the form of the equilibrated matrix.

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDA >= KL+KU+1.

           R

                     R is REAL array, dimension (M)
                     The row scale factors for A.

           C

                     C is REAL array, dimension (N)
                     The column scale factors for A.

           ROWCND

                     ROWCND is REAL
                     Ratio of the smallest R(i) to the largest R(i).

           COLCND

                     COLCND is REAL
                     Ratio of the smallest C(i) to the largest C(i).

           AMAX

                     AMAX is REAL
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED is CHARACTER*1
                     Specifies the form of equilibration that was done.
                     = 'N':  No equilibration
                     = 'R':  Row equilibration, i.e., A has been premultiplied by
                             diag(R).
                     = 'C':  Column equilibration, i.e., A has been postmultiplied
                             by diag(C).
                     = 'B':  Both row and column equilibration, i.e., A has been
                             replaced by diag(R) * A * diag(C).

       Internal Parameters:

             THRESH is a threshold value used to decide if row or column scaling
             should be done based on the ratio of the row or column scaling
             factors.  If ROWCND < THRESH, row scaling is done, and if
             COLCND < THRESH, column scaling is done.

             LARGE and SMALL are threshold values used to decide if row scaling
             should be done based on the absolute size of the largest matrix
             element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

Author

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