complexGBauxiliary(3) LAPACK complexGBauxiliary(3)

# NAME¶

complexGBauxiliary

# SYNOPSIS¶

## Functions¶

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

# Detailed Description¶

This is the group of complex auxiliary functions for GB matrices

# Function Documentation¶

## real function clangb (character NORM, integer N, integer KL, integer KU, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)¶

CLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.

Purpose:

``` CLANGB  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:

CLANGB

```    CLANGB = ( 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 CLANGB as described
above.
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.  When N = 0, CLANGB 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 COMPLEX 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 claqgb (integer M, integer N, integer KL, integer KU, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character EQUED)¶

CLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.

Purpose:

``` CLAQGB 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 COMPLEX 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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