LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
slansy.f
Go to the documentation of this file.
1 *> \brief \b SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SLANSY + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slansy.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slansy.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slansy.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * REAL FUNCTION SLANSY( NORM, UPLO, N, A, LDA, WORK )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER NORM, UPLO
25 * INTEGER LDA, N
26 * ..
27 * .. Array Arguments ..
28 * REAL A( LDA, * ), WORK( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SLANSY returns the value of the one norm, or the Frobenius norm, or
38 *> the infinity norm, or the element of largest absolute value of a
39 *> real symmetric matrix A.
40 *> \endverbatim
41 *>
42 *> \return SLANSY
43 *> \verbatim
44 *>
45 *> SLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
46 *> (
47 *> ( norm1(A), NORM = '1', 'O' or 'o'
48 *> (
49 *> ( normI(A), NORM = 'I' or 'i'
50 *> (
51 *> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
52 *>
53 *> where norm1 denotes the one norm of a matrix (maximum column sum),
54 *> normI denotes the infinity norm of a matrix (maximum row sum) and
55 *> normF denotes the Frobenius norm of a matrix (square root of sum of
56 *> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
57 *> \endverbatim
58 *
59 * Arguments:
60 * ==========
61 *
62 *> \param[in] NORM
63 *> \verbatim
64 *> NORM is CHARACTER*1
65 *> Specifies the value to be returned in SLANSY as described
66 *> above.
67 *> \endverbatim
68 *>
69 *> \param[in] UPLO
70 *> \verbatim
71 *> UPLO is CHARACTER*1
72 *> Specifies whether the upper or lower triangular part of the
73 *> symmetric matrix A is to be referenced.
74 *> = 'U': Upper triangular part of A is referenced
75 *> = 'L': Lower triangular part of A is referenced
76 *> \endverbatim
77 *>
78 *> \param[in] N
79 *> \verbatim
80 *> N is INTEGER
81 *> The order of the matrix A. N >= 0. When N = 0, SLANSY is
82 *> set to zero.
83 *> \endverbatim
84 *>
85 *> \param[in] A
86 *> \verbatim
87 *> A is REAL array, dimension (LDA,N)
88 *> The symmetric matrix A. If UPLO = 'U', the leading n by n
89 *> upper triangular part of A contains the upper triangular part
90 *> of the matrix A, and the strictly lower triangular part of A
91 *> is not referenced. If UPLO = 'L', the leading n by n lower
92 *> triangular part of A contains the lower triangular part of
93 *> the matrix A, and the strictly upper triangular part of A is
94 *> not referenced.
95 *> \endverbatim
96 *>
97 *> \param[in] LDA
98 *> \verbatim
99 *> LDA is INTEGER
100 *> The leading dimension of the array A. LDA >= max(N,1).
101 *> \endverbatim
102 *>
103 *> \param[out] WORK
104 *> \verbatim
105 *> WORK is REAL array, dimension (MAX(1,LWORK)),
106 *> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
107 *> WORK is not referenced.
108 *> \endverbatim
109 *
110 * Authors:
111 * ========
112 *
113 *> \author Univ. of Tennessee
114 *> \author Univ. of California Berkeley
115 *> \author Univ. of Colorado Denver
116 *> \author NAG Ltd.
117 *
118 *> \ingroup realSYauxiliary
119 *
120 * =====================================================================
121  REAL function slansy( norm, uplo, n, a, lda, work )
122 *
123 * -- LAPACK auxiliary routine --
124 * -- LAPACK is a software package provided by Univ. of Tennessee, --
125 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
126 *
127  IMPLICIT NONE
128 * .. Scalar Arguments ..
129  CHARACTER norm, uplo
130  INTEGER lda, n
131 * ..
132 * .. Array Arguments ..
133  REAL a( lda, * ), work( * )
134 * ..
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139  REAL one, zero
140  parameter( one = 1.0e+0, zero = 0.0e+0 )
141 * ..
142 * .. Local Scalars ..
143  INTEGER i, j
144  REAL absa, sum, value
145 * ..
146 * .. Local Arrays ..
147  REAL ssq( 2 ), colssq( 2 )
148 * ..
149 * .. External Functions ..
150  LOGICAL lsame, sisnan
151  EXTERNAL lsame, sisnan
152 * ..
153 * .. External Subroutines ..
154  EXTERNAL slassq, scombssq
155 * ..
156 * .. Intrinsic Functions ..
157  INTRINSIC abs, sqrt
158 * ..
159 * .. Executable Statements ..
160 *
161  IF( n.EQ.0 ) THEN
162  VALUE = zero
163  ELSE IF( lsame( norm, 'M' ) ) THEN
164 *
165 * Find max(abs(A(i,j))).
166 *
167  VALUE = zero
168  IF( lsame( uplo, 'U' ) ) THEN
169  DO 20 j = 1, n
170  DO 10 i = 1, j
171  sum = abs( a( i, j ) )
172  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
173  10 CONTINUE
174  20 CONTINUE
175  ELSE
176  DO 40 j = 1, n
177  DO 30 i = j, n
178  sum = abs( a( i, j ) )
179  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
180  30 CONTINUE
181  40 CONTINUE
182  END IF
183  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
184  $ ( norm.EQ.'1' ) ) THEN
185 *
186 * Find normI(A) ( = norm1(A), since A is symmetric).
187 *
188  VALUE = zero
189  IF( lsame( uplo, 'U' ) ) THEN
190  DO 60 j = 1, n
191  sum = zero
192  DO 50 i = 1, j - 1
193  absa = abs( a( i, j ) )
194  sum = sum + absa
195  work( i ) = work( i ) + absa
196  50 CONTINUE
197  work( j ) = sum + abs( a( j, j ) )
198  60 CONTINUE
199  DO 70 i = 1, n
200  sum = work( i )
201  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
202  70 CONTINUE
203  ELSE
204  DO 80 i = 1, n
205  work( i ) = zero
206  80 CONTINUE
207  DO 100 j = 1, n
208  sum = work( j ) + abs( a( j, j ) )
209  DO 90 i = j + 1, n
210  absa = abs( a( i, j ) )
211  sum = sum + absa
212  work( i ) = work( i ) + absa
213  90 CONTINUE
214  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
215  100 CONTINUE
216  END IF
217  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
218 *
219 * Find normF(A).
220 * SSQ(1) is scale
221 * SSQ(2) is sum-of-squares
222 * For better accuracy, sum each column separately.
223 *
224  ssq( 1 ) = zero
225  ssq( 2 ) = one
226 *
227 * Sum off-diagonals
228 *
229  IF( lsame( uplo, 'U' ) ) THEN
230  DO 110 j = 2, n
231  colssq( 1 ) = zero
232  colssq( 2 ) = one
233  CALL slassq( j-1, a( 1, j ), 1, colssq(1), colssq(2) )
234  CALL scombssq( ssq, colssq )
235  110 CONTINUE
236  ELSE
237  DO 120 j = 1, n - 1
238  colssq( 1 ) = zero
239  colssq( 2 ) = one
240  CALL slassq( n-j, a( j+1, j ), 1, colssq(1), colssq(2) )
241  CALL scombssq( ssq, colssq )
242  120 CONTINUE
243  END IF
244  ssq( 2 ) = 2*ssq( 2 )
245 *
246 * Sum diagonal
247 *
248  colssq( 1 ) = zero
249  colssq( 2 ) = one
250  CALL slassq( n, a, lda+1, colssq( 1 ), colssq( 2 ) )
251  CALL scombssq( ssq, colssq )
252  VALUE = ssq( 1 )*sqrt( ssq( 2 ) )
253  END IF
254 *
255  slansy = VALUE
256  RETURN
257 *
258 * End of SLANSY
259 *
260  END
subroutine slassq(n, x, incx, scl, sumsq)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f90:126
subroutine scombssq(V1, V2)
SCOMBSSQ adds two scaled sum of squares quantities
Definition: scombssq.f:60
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:59
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122