LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
dlansb.f
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1 *> \brief \b 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.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DLANSB + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansb.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
22 * WORK )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER NORM, UPLO
26 * INTEGER K, LDAB, N
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION AB( LDAB, * ), WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> DLANSB returns the value of the one norm, or the Frobenius norm, or
39 *> the infinity norm, or the element of largest absolute value of an
40 *> n by n symmetric band matrix A, with k super-diagonals.
41 *> \endverbatim
42 *>
43 *> \return DLANSB
44 *> \verbatim
45 *>
46 *> DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
47 *> (
48 *> ( norm1(A), NORM = '1', 'O' or 'o'
49 *> (
50 *> ( normI(A), NORM = 'I' or 'i'
51 *> (
52 *> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
53 *>
54 *> where norm1 denotes the one norm of a matrix (maximum column sum),
55 *> normI denotes the infinity norm of a matrix (maximum row sum) and
56 *> normF denotes the Frobenius norm of a matrix (square root of sum of
57 *> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
58 *> \endverbatim
59 *
60 * Arguments:
61 * ==========
62 *
63 *> \param[in] NORM
64 *> \verbatim
65 *> NORM is CHARACTER*1
66 *> Specifies the value to be returned in DLANSB as described
67 *> above.
68 *> \endverbatim
69 *>
70 *> \param[in] UPLO
71 *> \verbatim
72 *> UPLO is CHARACTER*1
73 *> Specifies whether the upper or lower triangular part of the
74 *> band matrix A is supplied.
75 *> = 'U': Upper triangular part is supplied
76 *> = 'L': Lower triangular part is supplied
77 *> \endverbatim
78 *>
79 *> \param[in] N
80 *> \verbatim
81 *> N is INTEGER
82 *> The order of the matrix A. N >= 0. When N = 0, DLANSB is
83 *> set to zero.
84 *> \endverbatim
85 *>
86 *> \param[in] K
87 *> \verbatim
88 *> K is INTEGER
89 *> The number of super-diagonals or sub-diagonals of the
90 *> band matrix A. K >= 0.
91 *> \endverbatim
92 *>
93 *> \param[in] AB
94 *> \verbatim
95 *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
96 *> The upper or lower triangle of the symmetric band matrix A,
97 *> stored in the first K+1 rows of AB. The j-th column of A is
98 *> stored in the j-th column of the array AB as follows:
99 *> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
100 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
101 *> \endverbatim
102 *>
103 *> \param[in] LDAB
104 *> \verbatim
105 *> LDAB is INTEGER
106 *> The leading dimension of the array AB. LDAB >= K+1.
107 *> \endverbatim
108 *>
109 *> \param[out] WORK
110 *> \verbatim
111 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
112 *> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
113 *> WORK is not referenced.
114 *> \endverbatim
115 *
116 * Authors:
117 * ========
118 *
119 *> \author Univ. of Tennessee
120 *> \author Univ. of California Berkeley
121 *> \author Univ. of Colorado Denver
122 *> \author NAG Ltd.
123 *
124 *> \date December 2016
125 *
126 *> \ingroup doubleOTHERauxiliary
127 *
128 * =====================================================================
129  DOUBLE PRECISION FUNCTION dlansb( NORM, UPLO, N, K, AB, LDAB,
130  $ WORK )
131 *
132 * -- LAPACK auxiliary routine (version 3.7.0) --
133 * -- LAPACK is a software package provided by Univ. of Tennessee, --
134 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
135 * December 2016
136 *
137 * .. Scalar Arguments ..
138  CHARACTER NORM, UPLO
139  INTEGER K, LDAB, N
140 * ..
141 * .. Array Arguments ..
142  DOUBLE PRECISION AB( ldab, * ), WORK( * )
143 * ..
144 *
145 * =====================================================================
146 *
147 * .. Parameters ..
148  DOUBLE PRECISION ONE, ZERO
149  parameter( one = 1.0d+0, zero = 0.0d+0 )
150 * ..
151 * .. Local Scalars ..
152  INTEGER I, J, L
153  DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
154 * ..
155 * .. External Subroutines ..
156  EXTERNAL dlassq
157 * ..
158 * .. External Functions ..
159  LOGICAL LSAME, DISNAN
160  EXTERNAL lsame, disnan
161 * ..
162 * .. Intrinsic Functions ..
163  INTRINSIC abs, max, min, sqrt
164 * ..
165 * .. Executable Statements ..
166 *
167  IF( n.EQ.0 ) THEN
168  VALUE = zero
169  ELSE IF( lsame( norm, 'M' ) ) THEN
170 *
171 * Find max(abs(A(i,j))).
172 *
173  VALUE = zero
174  IF( lsame( uplo, 'U' ) ) THEN
175  DO 20 j = 1, n
176  DO 10 i = max( k+2-j, 1 ), k + 1
177  sum = abs( ab( i, j ) )
178  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
179  10 CONTINUE
180  20 CONTINUE
181  ELSE
182  DO 40 j = 1, n
183  DO 30 i = 1, min( n+1-j, k+1 )
184  sum = abs( ab( i, j ) )
185  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
186  30 CONTINUE
187  40 CONTINUE
188  END IF
189  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
190  $ ( norm.EQ.'1' ) ) THEN
191 *
192 * Find normI(A) ( = norm1(A), since A is symmetric).
193 *
194  VALUE = zero
195  IF( lsame( uplo, 'U' ) ) THEN
196  DO 60 j = 1, n
197  sum = zero
198  l = k + 1 - j
199  DO 50 i = max( 1, j-k ), j - 1
200  absa = abs( ab( l+i, j ) )
201  sum = sum + absa
202  work( i ) = work( i ) + absa
203  50 CONTINUE
204  work( j ) = sum + abs( ab( k+1, j ) )
205  60 CONTINUE
206  DO 70 i = 1, n
207  sum = work( i )
208  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
209  70 CONTINUE
210  ELSE
211  DO 80 i = 1, n
212  work( i ) = zero
213  80 CONTINUE
214  DO 100 j = 1, n
215  sum = work( j ) + abs( ab( 1, j ) )
216  l = 1 - j
217  DO 90 i = j + 1, min( n, j+k )
218  absa = abs( ab( l+i, j ) )
219  sum = sum + absa
220  work( i ) = work( i ) + absa
221  90 CONTINUE
222  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
223  100 CONTINUE
224  END IF
225  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
226 *
227 * Find normF(A).
228 *
229  scale = zero
230  sum = one
231  IF( k.GT.0 ) THEN
232  IF( lsame( uplo, 'U' ) ) THEN
233  DO 110 j = 2, n
234  CALL dlassq( min( j-1, k ), ab( max( k+2-j, 1 ), j ),
235  $ 1, scale, sum )
236  110 CONTINUE
237  l = k + 1
238  ELSE
239  DO 120 j = 1, n - 1
240  CALL dlassq( min( n-j, k ), ab( 2, j ), 1, scale,
241  $ sum )
242  120 CONTINUE
243  l = 1
244  END IF
245  sum = 2*sum
246  ELSE
247  l = 1
248  END IF
249  CALL dlassq( n, ab( l, 1 ), ldab, scale, sum )
250  VALUE = scale*sqrt( sum )
251  END IF
252 *
253  dlansb = VALUE
254  RETURN
255 *
256 * End of DLANSB
257 *
258  END
subroutine dlassq(N, X, INCX, SCALE, SUMSQ)
DLASSQ updates a sum of squares represented in scaled form.
Definition: dlassq.f:105
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, or the element of largest absolute value of a symmetric band matrix.
Definition: dlansb.f:131