 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dlansb()

 double precision function dlansb ( character NORM, character UPLO, integer N, integer K, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK )

DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

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Purpose:
``` DLANSB  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 symmetric band matrix A,  with k super-diagonals.```
Returns
DLANSB
```    DLANSB = ( 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 DLANSB as described above.``` [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular part is supplied = 'L': Lower triangular part is supplied``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANSB is set to zero.``` [in] K ``` K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.``` [in] AB ``` AB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangle of the symmetric band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.```

Definition at line 127 of file dlansb.f.

129 *
130 * -- LAPACK auxiliary routine --
131 * -- LAPACK is a software package provided by Univ. of Tennessee, --
132 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133 *
134 * .. Scalar Arguments ..
135  CHARACTER NORM, UPLO
136  INTEGER K, LDAB, N
137 * ..
138 * .. Array Arguments ..
139  DOUBLE PRECISION AB( LDAB, * ), WORK( * )
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, L
150  DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL dlassq
154 * ..
155 * .. External Functions ..
156  LOGICAL LSAME, DISNAN
157  EXTERNAL lsame, disnan
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  IF( lsame( uplo, 'U' ) ) THEN
172  DO 20 j = 1, n
173  DO 10 i = max( k+2-j, 1 ), k + 1
174  sum = abs( ab( i, j ) )
175  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
176  10 CONTINUE
177  20 CONTINUE
178  ELSE
179  DO 40 j = 1, n
180  DO 30 i = 1, min( n+1-j, k+1 )
181  sum = abs( ab( i, j ) )
182  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
183  30 CONTINUE
184  40 CONTINUE
185  END IF
186  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
187  \$ ( norm.EQ.'1' ) ) THEN
188 *
189 * Find normI(A) ( = norm1(A), since A is symmetric).
190 *
191  VALUE = zero
192  IF( lsame( uplo, 'U' ) ) THEN
193  DO 60 j = 1, n
194  sum = zero
195  l = k + 1 - j
196  DO 50 i = max( 1, j-k ), j - 1
197  absa = abs( ab( l+i, j ) )
198  sum = sum + absa
199  work( i ) = work( i ) + absa
200  50 CONTINUE
201  work( j ) = sum + abs( ab( k+1, j ) )
202  60 CONTINUE
203  DO 70 i = 1, n
204  sum = work( i )
205  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
206  70 CONTINUE
207  ELSE
208  DO 80 i = 1, n
209  work( i ) = zero
210  80 CONTINUE
211  DO 100 j = 1, n
212  sum = work( j ) + abs( ab( 1, j ) )
213  l = 1 - j
214  DO 90 i = j + 1, min( n, j+k )
215  absa = abs( ab( l+i, j ) )
216  sum = sum + absa
217  work( i ) = work( i ) + absa
218  90 CONTINUE
219  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
220  100 CONTINUE
221  END IF
222  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
223 *
224 * Find normF(A).
225 *
226  scale = zero
227  sum = one
228  IF( k.GT.0 ) THEN
229  IF( lsame( uplo, 'U' ) ) THEN
230  DO 110 j = 2, n
231  CALL dlassq( min( j-1, k ), ab( max( k+2-j, 1 ), j ),
232  \$ 1, scale, sum )
233  110 CONTINUE
234  l = k + 1
235  ELSE
236  DO 120 j = 1, n - 1
237  CALL dlassq( min( n-j, k ), ab( 2, j ), 1, scale,
238  \$ sum )
239  120 CONTINUE
240  l = 1
241  END IF
242  sum = 2*sum
243  ELSE
244  l = 1
245  END IF
246  CALL dlassq( n, ab( l, 1 ), ldab, scale, sum )
247  VALUE = scale*sqrt( sum )
248  END IF
249 *
250  dlansb = VALUE
251  RETURN
252 *
253 * End of DLANSB
254 *
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:59
subroutine dlassq(n, x, incx, scl, sumsq)
DLASSQ updates a sum of squares represented in scaled form.
Definition: dlassq.f90:137
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dlansb(NORM, UPLO, N, K, AB, LDAB, WORK)
DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansb.f:129
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