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LAPACK » General Band Matrix » Auxiliary routines

Functions

double precision function dlangb (norm, n, kl, ku, ab, ldab, work)
 DLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.
subroutine dlaqgb (m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed)
 DLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ.

Detailed Description

This is the group of double auxiliary functions for GB matrices

Function Documentation

◆ dlangb()

double precision function dlangb ( character norm,
integer n,
integer kl,
integer ku,
double precision, dimension( ldab, * ) ab,
integer ldab,
double precision, dimension( * ) work )

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

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

Purpose:
!>
!> DLANGB  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
DLANGB 
!>
!>    DLANGB = ( 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 DLANGB as described
!>          above.
!>
[in]N
!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.  When N = 0, DLANGB 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 122 of file dlangb.f.

124*
125* -- LAPACK auxiliary routine --
126* -- LAPACK is a software package provided by Univ. of Tennessee, --
127* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
128*
129* .. Scalar Arguments ..
130 CHARACTER NORM
131 INTEGER KL, KU, LDAB, N
132* ..
133* .. Array Arguments ..
134 DOUBLE PRECISION AB( LDAB, * ), WORK( * )
135* ..
136*
137* =====================================================================
138*
139*
140* .. Parameters ..
141 DOUBLE PRECISION ONE, ZERO
142 parameter( one = 1.0d+0, zero = 0.0d+0 )
143* ..
144* .. Local Scalars ..
145 INTEGER I, J, K, L
146 DOUBLE PRECISION SCALE, SUM, VALUE, TEMP
147* ..
148* .. External Subroutines ..
149 EXTERNAL dlassq
150* ..
151* .. External Functions ..
152 LOGICAL LSAME, DISNAN
153 EXTERNAL lsame, disnan
154* ..
155* .. Intrinsic Functions ..
156 INTRINSIC abs, max, min, sqrt
157* ..
158* .. Executable Statements ..
159*
160 IF( n.EQ.0 ) THEN
161 VALUE = zero
162 ELSE IF( lsame( norm, 'M' ) ) THEN
163*
164* Find max(abs(A(i,j))).
165*
166 VALUE = zero
167 DO 20 j = 1, n
168 DO 10 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
169 temp = abs( ab( i, j ) )
170 IF( VALUE.LT.temp .OR. disnan( temp ) ) VALUE = temp
171 10 CONTINUE
172 20 CONTINUE
173 ELSE IF( ( lsame( norm, 'O' ) ) .OR. ( norm.EQ.'1' ) ) THEN
174*
175* Find norm1(A).
176*
177 VALUE = zero
178 DO 40 j = 1, n
179 sum = zero
180 DO 30 i = max( ku+2-j, 1 ), min( n+ku+1-j, kl+ku+1 )
181 sum = sum + abs( ab( i, j ) )
182 30 CONTINUE
183 IF( VALUE.LT.sum .OR. disnan( sum ) ) VALUE = sum
184 40 CONTINUE
185 ELSE IF( lsame( norm, 'I' ) ) THEN
186*
187* Find normI(A).
188*
189 DO 50 i = 1, n
190 work( i ) = zero
191 50 CONTINUE
192 DO 70 j = 1, n
193 k = ku + 1 - j
194 DO 60 i = max( 1, j-ku ), min( n, j+kl )
195 work( i ) = work( i ) + abs( ab( k+i, j ) )
196 60 CONTINUE
197 70 CONTINUE
198 VALUE = zero
199 DO 80 i = 1, n
200 temp = work( i )
201 IF( VALUE.LT.temp .OR. disnan( temp ) ) VALUE = temp
202 80 CONTINUE
203 ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
204*
205* Find normF(A).
206*
207 scale = zero
208 sum = one
209 DO 90 j = 1, n
210 l = max( 1, j-ku )
211 k = ku + 1 - j + l
212 CALL dlassq( min( n, j+kl )-l+1, ab( k, j ), 1, scale, sum )
213 90 CONTINUE
214 VALUE = scale*sqrt( sum )
215 END IF
216*
217 dlangb = VALUE
218 RETURN
219*
220* End of DLANGB
221*
norm
norm(diag(diag(diag(inv(mat))) -id.SOL), 2) % destroy mumps instance id.JOB
dlassq
subroutine dlassq(n, x, incx, scl, sumsq)
DLASSQ updates a sum of squares represented in scaled form.
Definition dlassq.f90:137
disnan
logical function disnan(din)
DISNAN tests input for NaN.
Definition disnan.f:59
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:53
dlangb
double precision function dlangb(norm, n, kl, ku, ab, ldab, work)
DLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition dlangb.f:124
min
#define min(a, b)
Definition macros.h:20
max
#define max(a, b)
Definition macros.h:21

◆ dlaqgb()

subroutine dlaqgb ( integer m,
integer n,
integer kl,
integer ku,
double precision, dimension( ldab, * ) ab,
integer ldab,
double precision, dimension( * ) r,
double precision, dimension( * ) c,
double precision rowcnd,
double precision colcnd,
double precision amax,
character equed )

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

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

Purpose:
!>
!> DLAQGB 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (M)
!>          The row scale factors for A.
!>
[in]C
!>          C is DOUBLE PRECISION array, dimension (N)
!>          The column scale factors for A.
!>
[in]ROWCND
!>          ROWCND is DOUBLE PRECISION
!>          Ratio of the smallest R(i) to the largest R(i).
!>
[in]COLCND
!>          COLCND is DOUBLE PRECISION
!>          Ratio of the smallest C(i) to the largest C(i).
!>
[in]AMAX
!>          AMAX is DOUBLE PRECISION
!>          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 157 of file dlaqgb.f.

159*
160* -- LAPACK auxiliary routine --
161* -- LAPACK is a software package provided by Univ. of Tennessee, --
162* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
163*
164* .. Scalar Arguments ..
165 CHARACTER EQUED
166 INTEGER KL, KU, LDAB, M, N
167 DOUBLE PRECISION AMAX, COLCND, ROWCND
168* ..
169* .. Array Arguments ..
170 DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
171* ..
172*
173* =====================================================================
174*
175* .. Parameters ..
176 DOUBLE PRECISION ONE, THRESH
177 parameter( one = 1.0d+0, thresh = 0.1d+0 )
178* ..
179* .. Local Scalars ..
180 INTEGER I, J
181 DOUBLE PRECISION CJ, LARGE, SMALL
182* ..
183* .. External Functions ..
184 DOUBLE PRECISION DLAMCH
185 EXTERNAL dlamch
186* ..
187* .. Intrinsic Functions ..
188 INTRINSIC max, min
189* ..
190* .. Executable Statements ..
191*
192* Quick return if possible
193*
194 IF( m.LE.0 .OR. n.LE.0 ) THEN
195 equed = 'N'
196 RETURN
197 END IF
198*
199* Initialize LARGE and SMALL.
200*
201 small = dlamch( 'Safe minimum' ) / dlamch( 'Precision' )
202 large = one / small
203*
204 IF( rowcnd.GE.thresh .AND. amax.GE.small .AND. amax.LE.large )
205 $ THEN
206*
207* No row scaling
208*
209 IF( colcnd.GE.thresh ) THEN
210*
211* No column scaling
212*
213 equed = 'N'
214 ELSE
215*
216* Column scaling
217*
218 DO 20 j = 1, n
219 cj = c( j )
220 DO 10 i = max( 1, j-ku ), min( m, j+kl )
221 ab( ku+1+i-j, j ) = cj*ab( ku+1+i-j, j )
222 10 CONTINUE
223 20 CONTINUE
224 equed = 'C'
225 END IF
226 ELSE IF( colcnd.GE.thresh ) THEN
227*
228* Row scaling, no column scaling
229*
230 DO 40 j = 1, n
231 DO 30 i = max( 1, j-ku ), min( m, j+kl )
232 ab( ku+1+i-j, j ) = r( i )*ab( ku+1+i-j, j )
233 30 CONTINUE
234 40 CONTINUE
235 equed = 'R'
236 ELSE
237*
238* Row and column scaling
239*
240 DO 60 j = 1, n
241 cj = c( j )
242 DO 50 i = max( 1, j-ku ), min( m, j+kl )
243 ab( ku+1+i-j, j ) = cj*r( i )*ab( ku+1+i-j, j )
244 50 CONTINUE
245 60 CONTINUE
246 equed = 'B'
247 END IF
248*
249 RETURN
250*
251* End of DLAQGB
252*
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69