LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
Collaboration diagram for complex16:

Functions

double precision function zlangb (NORM, N, KL, KU, AB, LDAB, WORK)
 ZLANGB 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 zlaqgb (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, EQUED)
 ZLAQGB scales a general band matrix, using row and column scaling factors computed by sgbequ. More...
 

Detailed Description

This is the group of complex16 auxiliary functions for GB matrices

Function Documentation

double precision function zlangb ( character  NORM,
integer  N,
integer  KL,
integer  KU,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  WORK 
)

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

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

Purpose:
 ZLANGB  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
ZLANGB
    ZLANGB = ( 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 ZLANGB as described
          above.
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, ZLANGB 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*16 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.
Date
September 2012

Definition at line 127 of file zlangb.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  DOUBLE PRECISION work( * )
139  COMPLEX*16 ab( ldab, * )
140 * ..
141 *
142 * =====================================================================
143 *
144 * .. Parameters ..
145  DOUBLE PRECISION one, zero
146  parameter( one = 1.0d+0, zero = 0.0d+0 )
147 * ..
148 * .. Local Scalars ..
149  INTEGER i, j, k, l
150  DOUBLE PRECISION scale, sum, VALUE, temp
151 * ..
152 * .. External Functions ..
153  LOGICAL lsame, disnan
154  EXTERNAL lsame, disnan
155 * ..
156 * .. External Subroutines ..
157  EXTERNAL zlassq
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. disnan( 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. disnan( 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. disnan( 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 zlassq( 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  zlangb = VALUE
222  RETURN
223 *
224 * End of ZLANGB
225 *
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:61
subroutine zlassq(N, X, INCX, SCALE, SUMSQ)
ZLASSQ updates a sum of squares represented in scaled form.
Definition: zlassq.f:108
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
double precision function zlangb(NORM, N, KL, KU, AB, LDAB, WORK)
ZLANGB returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlangb.f:127

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subroutine zlaqgb ( integer  M,
integer  N,
integer  KL,
integer  KU,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
double precision  ROWCND,
double precision  COLCND,
double precision  AMAX,
character  EQUED 
)

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

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

Purpose:
 ZLAQGB 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*16 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.
Date
September 2012

Definition at line 162 of file zlaqgb.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  DOUBLE PRECISION amax, colcnd, rowcnd
172 * ..
173 * .. Array Arguments ..
174  DOUBLE PRECISION c( * ), r( * )
175  COMPLEX*16 ab( ldab, * )
176 * ..
177 *
178 * =====================================================================
179 *
180 * .. Parameters ..
181  DOUBLE PRECISION one, thresh
182  parameter( one = 1.0d+0, thresh = 0.1d+0 )
183 * ..
184 * .. Local Scalars ..
185  INTEGER i, j
186  DOUBLE PRECISION cj, large, small
187 * ..
188 * .. External Functions ..
189  DOUBLE PRECISION dlamch
190  EXTERNAL dlamch
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 = dlamch( 'Safe minimum' ) / dlamch( '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 ZLAQGB
257 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65

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