man realGBauxiliary (3): real

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

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

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

          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:

: September 2012

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

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

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

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

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

          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.