LAPACK  3.9.0 LAPACK: Linear Algebra PACKage
clanhe.f
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1 *> \brief \b CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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17 *
18 * Definition:
19 * ===========
20 *
21 * REAL FUNCTION CLANHE( NORM, UPLO, N, A, LDA, WORK )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER NORM, UPLO
25 * INTEGER LDA, N
26 * ..
27 * .. Array Arguments ..
28 * REAL WORK( * )
29 * COMPLEX A( LDA, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CLANHE returns the value of the one norm, or the Frobenius norm, or
39 *> the infinity norm, or the element of largest absolute value of a
40 *> complex hermitian matrix A.
41 *> \endverbatim
42 *>
43 *> \return CLANHE
44 *> \verbatim
45 *>
46 *> CLANHE = ( 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 CLANHE 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 *> hermitian matrix A is to be referenced.
75 *> = 'U': Upper triangular part of A is referenced
76 *> = 'L': Lower triangular part of A is referenced
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, CLANHE is
83 *> set to zero.
84 *> \endverbatim
85 *>
86 *> \param[in] A
87 *> \verbatim
88 *> A is COMPLEX array, dimension (LDA,N)
89 *> The hermitian matrix A. If UPLO = 'U', the leading n by n
90 *> upper triangular part of A contains the upper triangular part
91 *> of the matrix A, and the strictly lower triangular part of A
92 *> is not referenced. If UPLO = 'L', the leading n by n lower
93 *> triangular part of A contains the lower triangular part of
94 *> the matrix A, and the strictly upper triangular part of A is
95 *> not referenced. Note that the imaginary parts of the diagonal
96 *> elements need not be set and are assumed to be zero.
97 *> \endverbatim
98 *>
99 *> \param[in] LDA
100 *> \verbatim
101 *> LDA is INTEGER
102 *> The leading dimension of the array A. LDA >= max(N,1).
103 *> \endverbatim
104 *>
105 *> \param[out] WORK
106 *> \verbatim
107 *> WORK is REAL array, dimension (MAX(1,LWORK)),
108 *> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
109 *> WORK is not referenced.
110 *> \endverbatim
111 *
112 * Authors:
113 * ========
114 *
115 *> \author Univ. of Tennessee
116 *> \author Univ. of California Berkeley
117 *> \author Univ. of Colorado Denver
118 *> \author NAG Ltd.
119 *
120 *> \date December 2016
121 *
122 *> \ingroup complexHEauxiliary
123 *
124 * =====================================================================
125  REAL FUNCTION CLANHE( NORM, UPLO, N, A, LDA, WORK )
126 *
127 * -- LAPACK auxiliary routine (version 3.7.0) --
128 * -- LAPACK is a software package provided by Univ. of Tennessee, --
129 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
130 * December 2016
131 *
132  IMPLICIT NONE
133 * .. Scalar Arguments ..
134  CHARACTER norm, uplo
135  INTEGER lda, n
136 * ..
137 * .. Array Arguments ..
138  REAL work( * )
139  COMPLEX a( lda, * )
140 * ..
141 *
142 * =====================================================================
143 *
144 * .. Parameters ..
145  REAL one, zero
146  parameter( one = 1.0e+0, zero = 0.0e+0 )
147 * ..
148 * .. Local Scalars ..
149  INTEGER i, j
150  REAL absa, sum, value
151 * ..
152 * .. Local Arrays ..
153  REAL ssq( 2 ), colssq( 2 )
154 * ..
155 * .. External Functions ..
156  LOGICAL lsame, sisnan
157  EXTERNAL lsame, sisnan
158 * ..
159 * .. External Subroutines ..
160  EXTERNAL classq, scombssq
161 * ..
162 * .. Intrinsic Functions ..
163  INTRINSIC abs, real, 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 = 1, j - 1
177  sum = abs( a( i, j ) )
178  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
179  10 CONTINUE
180  sum = abs( real( a( j, j ) ) )
181  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
182  20 CONTINUE
183  ELSE
184  DO 40 j = 1, n
185  sum = abs( real( a( j, j ) ) )
186  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
187  DO 30 i = j + 1, n
188  sum = abs( a( i, j ) )
189  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
190  30 CONTINUE
191  40 CONTINUE
192  END IF
193  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
194  \$ ( norm.EQ.'1' ) ) THEN
195 *
196 * Find normI(A) ( = norm1(A), since A is hermitian).
197 *
198  VALUE = zero
199  IF( lsame( uplo, 'U' ) ) THEN
200  DO 60 j = 1, n
201  sum = zero
202  DO 50 i = 1, j - 1
203  absa = abs( a( i, j ) )
204  sum = sum + absa
205  work( i ) = work( i ) + absa
206  50 CONTINUE
207  work( j ) = sum + abs( real( a( j, j ) ) )
208  60 CONTINUE
209  DO 70 i = 1, n
210  sum = work( i )
211  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
212  70 CONTINUE
213  ELSE
214  DO 80 i = 1, n
215  work( i ) = zero
216  80 CONTINUE
217  DO 100 j = 1, n
218  sum = work( j ) + abs( real( a( j, j ) ) )
219  DO 90 i = j + 1, n
220  absa = abs( a( i, j ) )
221  sum = sum + absa
222  work( i ) = work( i ) + absa
223  90 CONTINUE
224  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
225  100 CONTINUE
226  END IF
227  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
228 *
229 * Find normF(A).
230 * SSQ(1) is scale
231 * SSQ(2) is sum-of-squares
232 * For better accuracy, sum each column separately.
233 *
234  ssq( 1 ) = zero
235  ssq( 2 ) = one
236 *
237 * Sum off-diagonals
238 *
239  IF( lsame( uplo, 'U' ) ) THEN
240  DO 110 j = 2, n
241  colssq( 1 ) = zero
242  colssq( 2 ) = one
243  CALL classq( j-1, a( 1, j ), 1,
244  \$ colssq( 1 ), colssq( 2 ) )
245  CALL scombssq( ssq, colssq )
246  110 CONTINUE
247  ELSE
248  DO 120 j = 1, n - 1
249  colssq( 1 ) = zero
250  colssq( 2 ) = one
251  CALL classq( n-j, a( j+1, j ), 1,
252  \$ colssq( 1 ), colssq( 2 ) )
253  CALL scombssq( ssq, colssq )
254  120 CONTINUE
255  END IF
256  ssq( 2 ) = 2*ssq( 2 )
257 *
258 * Sum diagonal
259 *
260  DO 130 i = 1, n
261  IF( real( a( i, i ) ).NE.zero ) THEN
262  absa = abs( real( a( i, i ) ) )
263  IF( ssq( 1 ).LT.absa ) THEN
264  ssq( 2 ) = one + ssq( 2 )*( ssq( 1 ) / absa )**2
265  ssq( 1 ) = absa
266  ELSE
267  ssq( 2 ) = ssq( 2 ) + ( absa / ssq( 1 ) )**2
268  END IF
269  END IF
270  130 CONTINUE
271  VALUE = ssq( 1 )*sqrt( ssq( 2 ) )
272  END IF
273 *
274  clanhe = VALUE
275  RETURN
276 *
277 * End of CLANHE
278 *
279  END
classq
subroutine classq(N, X, INCX, SCALE, SUMSQ)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f:108
sisnan
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:61
clanhe
real function clanhe(NORM, UPLO, N, A, LDA, WORK)
CLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhe.f:126
lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
scombssq
subroutine scombssq(V1, V2)
SCOMBSSQ adds two scaled sum of squares quantities
Definition: scombssq.f:62