LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansb.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansb.f">
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*> \ingroup lanhb
125*
126* =====================================================================
127 DOUBLE PRECISION FUNCTION dlansb( NORM, UPLO, N, K, AB, LDAB,
128 \$ WORK )
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*
255 END
logical function disnan(din)
DISNAN tests input for NaN.
Definition disnan.f:59
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
subroutine dlassq(n, x, incx, scale, sumsq)
DLASSQ updates a sum of squares represented in scaled form.
Definition dlassq.f90:124
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48