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complex
LAPACK » General Band Matrix » Auxiliary routines

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

◆ clangb()

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.

Download CLANGB + dependencies [TGZ] [ZIP] [TXT]

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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 123 of file clangb.f.

125*
126* -- LAPACK auxiliary routine --
127* -- LAPACK is a software package provided by Univ. of Tennessee, --
128* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
129*
130* .. Scalar Arguments ..
131 CHARACTER NORM
132 INTEGER KL, KU, LDAB, N
133* ..
134* .. Array Arguments ..
135 REAL WORK( * )
136 COMPLEX AB( LDAB, * )
137* ..
138*
139* =====================================================================
140*
141* .. Parameters ..
142 REAL ONE, ZERO
143 parameter( one = 1.0e+0, zero = 0.0e+0 )
144* ..
145* .. Local Scalars ..
146 INTEGER I, J, K, L
147 REAL SCALE, SUM, VALUE, TEMP
148* ..
149* .. External Functions ..
150 LOGICAL LSAME, SISNAN
151 EXTERNAL lsame, sisnan
152* ..
153* .. External Subroutines ..
154 EXTERNAL classq
155* ..
156* .. Intrinsic Functions ..
157 INTRINSIC abs, max, min, sqrt
158* ..
159* .. Executable Statements ..
160*
161 IF( n.EQ.0 ) THEN
162 VALUE = zero
163 ELSE IF( lsame( norm, 'M' ) ) THEN
164*
165* Find max(abs(A(i,j))).
166*
167 VALUE = zero
168 DO 20 j = 1, n
169 DO 10 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
170 temp = abs( ab( i, j ) )
171 IF( VALUE.LT.temp .OR. sisnan( temp ) ) VALUE = temp
172 10 CONTINUE
173 20 CONTINUE
174 ELSE IF( ( lsame( norm, 'O' ) ) .OR. ( norm.EQ.'1' ) ) THEN
175*
176* Find norm1(A).
177*
178 VALUE = zero
179 DO 40 j = 1, n
180 sum = zero
181 DO 30 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
182 sum = sum + abs( ab( i, j ) )
183 30 CONTINUE
184 IF( VALUE.LT.sum .OR. sisnan( sum ) ) VALUE = sum
185 40 CONTINUE
186 ELSE IF( lsame( norm, 'I' ) ) THEN
187*
188* Find normI(A).
189*
190 DO 50 i = 1, n
191 work( i ) = zero
192 50 CONTINUE
193 DO 70 j = 1, n
194 k = ku + 1 - j
195 DO 60 i = max( 1, j-ku ), min( n, j+kl )
196 work( i ) = work( i ) + abs( ab( k+i, j ) )
197 60 CONTINUE
198 70 CONTINUE
199 VALUE = zero
200 DO 80 i = 1, n
201 temp = work( i )
202 IF( VALUE.LT.temp .OR. sisnan( temp ) ) VALUE = temp
203 80 CONTINUE
204 ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
205*
206* Find normF(A).
207*
208 scale = zero
209 sum = one
210 DO 90 j = 1, n
211 l = max( 1, j-ku )
212 k = ku + 1 - j + l
213 CALL classq( min( n, j+kl )-l+1, ab( k, j ), 1, scale, sum )
214 90 CONTINUE
215 VALUE = scale*sqrt( sum )
216 END IF
217*
218 clangb = VALUE
219 RETURN
220*
221* End of CLANGB
222*
norm
norm(diag(diag(diag(inv(mat))) -id.SOL), 2) % destroy mumps instance id.JOB
sisnan
logical function sisnan(sin)
SISNAN tests input for NaN.
Definition sisnan.f:59
classq
subroutine classq(n, x, incx, scl, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition classq.f90:137
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:53
clangb
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:125
min
#define min(a, b)
Definition macros.h:20
max
#define max(a, b)
Definition macros.h:21

◆ claqgb()

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.

Download CLAQGB + dependencies [TGZ] [ZIP] [TXT]

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.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 158 of file claqgb.f.

160*
161* -- LAPACK auxiliary routine --
162* -- LAPACK is a software package provided by Univ. of Tennessee, --
163* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
164*
165* .. Scalar Arguments ..
166 CHARACTER EQUED
167 INTEGER KL, KU, LDAB, M, N
168 REAL AMAX, COLCND, ROWCND
169* ..
170* .. Array Arguments ..
171 REAL C( * ), R( * )
172 COMPLEX AB( LDAB, * )
173* ..
174*
175* =====================================================================
176*
177* .. Parameters ..
178 REAL ONE, THRESH
179 parameter( one = 1.0e+0, thresh = 0.1e+0 )
180* ..
181* .. Local Scalars ..
182 INTEGER I, J
183 REAL CJ, LARGE, SMALL
184* ..
185* .. External Functions ..
186 REAL SLAMCH
187 EXTERNAL slamch
188* ..
189* .. Intrinsic Functions ..
190 INTRINSIC max, min
191* ..
192* .. Executable Statements ..
193*
194* Quick return if possible
195*
196 IF( m.LE.0 .OR. n.LE.0 ) THEN
197 equed = 'N'
198 RETURN
199 END IF
200*
201* Initialize LARGE and SMALL.
202*
203 small = slamch( 'Safe minimum' ) / slamch( 'Precision' )
204 large = one / small
205*
206 IF( rowcnd.GE.thresh .AND. amax.GE.small .AND. amax.LE.large )
207 $ THEN
208*
209* No row scaling
210*
211 IF( colcnd.GE.thresh ) THEN
212*
213* No column scaling
214*
215 equed = 'N'
216 ELSE
217*
218* Column scaling
219*
220 DO 20 j = 1, n
221 cj = c( j )
222 DO 10 i = max( 1, j-ku ), min( m, j+kl )
223 ab( ku+1+i-j, j ) = cj*ab( ku+1+i-j, j )
224 10 CONTINUE
225 20 CONTINUE
226 equed = 'C'
227 END IF
228 ELSE IF( colcnd.GE.thresh ) THEN
229*
230* Row scaling, no column scaling
231*
232 DO 40 j = 1, n
233 DO 30 i = max( 1, j-ku ), min( m, j+kl )
234 ab( ku+1+i-j, j ) = r( i )*ab( ku+1+i-j, j )
235 30 CONTINUE
236 40 CONTINUE
237 equed = 'R'
238 ELSE
239*
240* Row and column scaling
241*
242 DO 60 j = 1, n
243 cj = c( j )
244 DO 50 i = max( 1, j-ku ), min( m, j+kl )
245 ab( ku+1+i-j, j ) = cj*r( i )*ab( ku+1+i-j, j )
246 50 CONTINUE
247 60 CONTINUE
248 equed = 'B'
249 END IF
250*
251 RETURN
252*
253* End of CLAQGB
254*
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68