LAPACK  3.6.0 LAPACK: Linear Algebra PACKage
complex
Collaboration diagram for complex:


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## 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. More...

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. More...

## 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
 [in] NORM ``` NORM is CHARACTER*1 Specifies the value to be returned in CLANGB as described above.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0. When N = 0, CLANGB is set to zero.``` [in] KL ``` KL is INTEGER The number of sub-diagonals of the matrix A. KL >= 0.``` [in] KU ``` KU is INTEGER The number of super-diagonals of the matrix A. KU >= 0.``` [in] 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).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1.``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.```
Date
September 2012

Definition at line 127 of file clangb.f.

127 *
128 * -- LAPACK auxiliary routine (version 3.4.2) --
129 * -- LAPACK is a software package provided by Univ. of Tennessee, --
130 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
131 * September 2012
132 *
133 * .. Scalar Arguments ..
134  CHARACTER norm
135  INTEGER kl, ku, ldab, n
136 * ..
137 * .. Array Arguments ..
138  REAL work( * )
139  COMPLEX ab( ldab, * )
140 * ..
141 *
142 * =====================================================================
143 *
144 * .. Parameters ..
145  REAL one, zero
146  parameter( one = 1.0e+0, zero = 0.0e+0 )
147 * ..
148 * .. Local Scalars ..
149  INTEGER i, j, k, l
150  REAL scale, sum, VALUE, temp
151 * ..
152 * .. External Functions ..
153  LOGICAL lsame, sisnan
154  EXTERNAL lsame, sisnan
155 * ..
156 * .. External Subroutines ..
157  EXTERNAL classq
158 * ..
159 * .. Intrinsic Functions ..
160  INTRINSIC abs, max, min, sqrt
161 * ..
162 * .. Executable Statements ..
163 *
164  IF( n.EQ.0 ) THEN
165  VALUE = zero
166  ELSE IF( lsame( norm, 'M' ) ) THEN
167 *
168 * Find max(abs(A(i,j))).
169 *
170  VALUE = zero
171  DO 20 j = 1, n
172  DO 10 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
173  temp = abs( ab( i, j ) )
174  IF( VALUE.LT.temp .OR. sisnan( temp ) ) VALUE = temp
175  10 CONTINUE
176  20 CONTINUE
177  ELSE IF( ( lsame( norm, 'O' ) ) .OR. ( norm.EQ.'1' ) ) THEN
178 *
179 * Find norm1(A).
180 *
181  VALUE = zero
182  DO 40 j = 1, n
183  sum = zero
184  DO 30 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
185  sum = sum + abs( ab( i, j ) )
186  30 CONTINUE
187  IF( VALUE.LT.sum .OR. sisnan( sum ) ) VALUE = sum
188  40 CONTINUE
189  ELSE IF( lsame( norm, 'I' ) ) THEN
190 *
191 * Find normI(A).
192 *
193  DO 50 i = 1, n
194  work( i ) = zero
195  50 CONTINUE
196  DO 70 j = 1, n
197  k = ku + 1 - j
198  DO 60 i = max( 1, j-ku ), min( n, j+kl )
199  work( i ) = work( i ) + abs( ab( k+i, j ) )
200  60 CONTINUE
201  70 CONTINUE
202  VALUE = zero
203  DO 80 i = 1, n
204  temp = work( i )
205  IF( VALUE.LT.temp .OR. sisnan( temp ) ) VALUE = temp
206  80 CONTINUE
207  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
208 *
209 * Find normF(A).
210 *
211  scale = zero
212  sum = one
213  DO 90 j = 1, n
214  l = max( 1, j-ku )
215  k = ku + 1 - j + l
216  CALL classq( min( n, j+kl )-l+1, ab( k, j ), 1, scale, sum )
217  90 CONTINUE
218  VALUE = scale*sqrt( sum )
219  END IF
220 *
221  clangb = VALUE
222  RETURN
223 *
224 * End of CLANGB
225 *
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:61
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 ...
Definition: clangb.f:127
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine classq(N, X, INCX, SCALE, SUMSQ)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f:108

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 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
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] KL ``` KL is INTEGER The number of subdiagonals within the band of A. KL >= 0.``` [in] KU ``` KU is INTEGER The number of superdiagonals within the band of A. KU >= 0.``` [in,out] 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.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDA >= KL+KU+1.``` [in] R ``` R is REAL array, dimension (M) The row scale factors for A.``` [in] C ``` C is REAL array, dimension (N) The column scale factors for A.``` [in] ROWCND ``` ROWCND is REAL Ratio of the smallest R(i) to the largest R(i).``` [in] COLCND ``` COLCND is REAL Ratio of the smallest C(i) to the largest C(i).``` [in] AMAX ``` AMAX is REAL Absolute value of largest matrix entry.``` [out] 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.```
Date
September 2012

Definition at line 162 of file claqgb.f.

162 *
163 * -- LAPACK auxiliary routine (version 3.4.2) --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 * September 2012
167 *
168 * .. Scalar Arguments ..
169  CHARACTER equed
170  INTEGER kl, ku, ldab, m, n
171  REAL amax, colcnd, rowcnd
172 * ..
173 * .. Array Arguments ..
174  REAL c( * ), r( * )
175  COMPLEX ab( ldab, * )
176 * ..
177 *
178 * =====================================================================
179 *
180 * .. Parameters ..
181  REAL one, thresh
182  parameter( one = 1.0e+0, thresh = 0.1e+0 )
183 * ..
184 * .. Local Scalars ..
185  INTEGER i, j
186  REAL cj, large, small
187 * ..
188 * .. External Functions ..
189  REAL slamch
190  EXTERNAL slamch
191 * ..
192 * .. Intrinsic Functions ..
193  INTRINSIC max, min
194 * ..
195 * .. Executable Statements ..
196 *
197 * Quick return if possible
198 *
199  IF( m.LE.0 .OR. n.LE.0 ) THEN
200  equed = 'N'
201  RETURN
202  END IF
203 *
204 * Initialize LARGE and SMALL.
205 *
206  small = slamch( 'Safe minimum' ) / slamch( 'Precision' )
207  large = one / small
208 *
209  IF( rowcnd.GE.thresh .AND. amax.GE.small .AND. amax.LE.large )
210  \$ THEN
211 *
212 * No row scaling
213 *
214  IF( colcnd.GE.thresh ) THEN
215 *
216 * No column scaling
217 *
218  equed = 'N'
219  ELSE
220 *
221 * Column scaling
222 *
223  DO 20 j = 1, n
224  cj = c( j )
225  DO 10 i = max( 1, j-ku ), min( m, j+kl )
226  ab( ku+1+i-j, j ) = cj*ab( ku+1+i-j, j )
227  10 CONTINUE
228  20 CONTINUE
229  equed = 'C'
230  END IF
231  ELSE IF( colcnd.GE.thresh ) THEN
232 *
233 * Row scaling, no column scaling
234 *
235  DO 40 j = 1, n
236  DO 30 i = max( 1, j-ku ), min( m, j+kl )
237  ab( ku+1+i-j, j ) = r( i )*ab( ku+1+i-j, j )
238  30 CONTINUE
239  40 CONTINUE
240  equed = 'R'
241  ELSE
242 *
243 * Row and column scaling
244 *
245  DO 60 j = 1, n
246  cj = c( j )
247  DO 50 i = max( 1, j-ku ), min( m, j+kl )
248  ab( ku+1+i-j, j ) = cj*r( i )*ab( ku+1+i-j, j )
249  50 CONTINUE
250  60 CONTINUE
251  equed = 'B'
252  END IF
253 *
254  RETURN
255 *
256 * End of CLAQGB
257 *
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69

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