LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
clanhb.f
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1 *> \brief \b CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * REAL FUNCTION CLANHB( 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 * REAL WORK( * )
30 * COMPLEX AB( LDAB, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CLANHB returns the value of the one norm, or the Frobenius norm, or
40 *> the infinity norm, or the element of largest absolute value of an
41 *> n by n hermitian band matrix A, with k super-diagonals.
42 *> \endverbatim
43 *>
44 *> \return CLANHB
45 *> \verbatim
46 *>
47 *> CLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
48 *> (
49 *> ( norm1(A), NORM = '1', 'O' or 'o'
50 *> (
51 *> ( normI(A), NORM = 'I' or 'i'
52 *> (
53 *> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
54 *>
55 *> where norm1 denotes the one norm of a matrix (maximum column sum),
56 *> normI denotes the infinity norm of a matrix (maximum row sum) and
57 *> normF denotes the Frobenius norm of a matrix (square root of sum of
58 *> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
59 *> \endverbatim
60 *
61 * Arguments:
62 * ==========
63 *
64 *> \param[in] NORM
65 *> \verbatim
66 *> NORM is CHARACTER*1
67 *> Specifies the value to be returned in CLANHB as described
68 *> above.
69 *> \endverbatim
70 *>
71 *> \param[in] UPLO
72 *> \verbatim
73 *> UPLO is CHARACTER*1
74 *> Specifies whether the upper or lower triangular part of the
75 *> band matrix A is supplied.
76 *> = 'U': Upper triangular
77 *> = 'L': Lower triangular
78 *> \endverbatim
79 *>
80 *> \param[in] N
81 *> \verbatim
82 *> N is INTEGER
83 *> The order of the matrix A. N >= 0. When N = 0, CLANHB is
84 *> set to zero.
85 *> \endverbatim
86 *>
87 *> \param[in] K
88 *> \verbatim
89 *> K is INTEGER
90 *> The number of super-diagonals or sub-diagonals of the
91 *> band matrix A. K >= 0.
92 *> \endverbatim
93 *>
94 *> \param[in] AB
95 *> \verbatim
96 *> AB is COMPLEX array, dimension (LDAB,N)
97 *> The upper or lower triangle of the hermitian band matrix A,
98 *> stored in the first K+1 rows of AB. The j-th column of A is
99 *> stored in the j-th column of the array AB as follows:
100 *> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
101 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
102 *> Note that the imaginary parts of the diagonal elements need
103 *> not be set and are assumed to be zero.
104 *> \endverbatim
105 *>
106 *> \param[in] LDAB
107 *> \verbatim
108 *> LDAB is INTEGER
109 *> The leading dimension of the array AB. LDAB >= K+1.
110 *> \endverbatim
111 *>
112 *> \param[out] WORK
113 *> \verbatim
114 *> WORK is REAL array, dimension (MAX(1,LWORK)),
115 *> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
116 *> WORK is not referenced.
117 *> \endverbatim
118 *
119 * Authors:
120 * ========
121 *
122 *> \author Univ. of Tennessee
123 *> \author Univ. of California Berkeley
124 *> \author Univ. of Colorado Denver
125 *> \author NAG Ltd.
126 *
127 *> \ingroup complexOTHERauxiliary
128 *
129 * =====================================================================
130  REAL function clanhb( norm, uplo, n, k, ab, ldab,
131  \$ work )
132 *
133 * -- LAPACK auxiliary routine --
134 * -- LAPACK is a software package provided by Univ. of Tennessee, --
135 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136 *
137 * .. Scalar Arguments ..
138  CHARACTER norm, uplo
139  INTEGER k, ldab, n
140 * ..
141 * .. Array Arguments ..
142  REAL work( * )
143  COMPLEX ab( ldab, * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149  REAL one, zero
150  parameter( one = 1.0e+0, zero = 0.0e+0 )
151 * ..
152 * .. Local Scalars ..
153  INTEGER i, j, l
154  REAL absa, scale, sum, value
155 * ..
156 * .. External Functions ..
157  LOGICAL lsame, sisnan
158  EXTERNAL lsame, sisnan
159 * ..
160 * .. External Subroutines ..
161  EXTERNAL classq
162 * ..
163 * .. Intrinsic Functions ..
164  INTRINSIC abs, max, min, real, sqrt
165 * ..
166 * .. Executable Statements ..
167 *
168  IF( n.EQ.0 ) THEN
169  VALUE = zero
170  ELSE IF( lsame( norm, 'M' ) ) THEN
171 *
172 * Find max(abs(A(i,j))).
173 *
174  VALUE = zero
175  IF( lsame( uplo, 'U' ) ) THEN
176  DO 20 j = 1, n
177  DO 10 i = max( k+2-j, 1 ), k
178  sum = abs( ab( i, j ) )
179  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
180  10 CONTINUE
181  sum = abs( real( ab( k+1, j ) ) )
182  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
183  20 CONTINUE
184  ELSE
185  DO 40 j = 1, n
186  sum = abs( real( ab( 1, j ) ) )
187  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
188  DO 30 i = 2, min( n+1-j, k+1 )
189  sum = abs( ab( i, j ) )
190  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
191  30 CONTINUE
192  40 CONTINUE
193  END IF
194  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
195  \$ ( norm.EQ.'1' ) ) THEN
196 *
197 * Find normI(A) ( = norm1(A), since A is hermitian).
198 *
199  VALUE = zero
200  IF( lsame( uplo, 'U' ) ) THEN
201  DO 60 j = 1, n
202  sum = zero
203  l = k + 1 - j
204  DO 50 i = max( 1, j-k ), j - 1
205  absa = abs( ab( l+i, j ) )
206  sum = sum + absa
207  work( i ) = work( i ) + absa
208  50 CONTINUE
209  work( j ) = sum + abs( real( ab( k+1, j ) ) )
210  60 CONTINUE
211  DO 70 i = 1, n
212  sum = work( i )
213  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
214  70 CONTINUE
215  ELSE
216  DO 80 i = 1, n
217  work( i ) = zero
218  80 CONTINUE
219  DO 100 j = 1, n
220  sum = work( j ) + abs( real( ab( 1, j ) ) )
221  l = 1 - j
222  DO 90 i = j + 1, min( n, j+k )
223  absa = abs( ab( l+i, j ) )
224  sum = sum + absa
225  work( i ) = work( i ) + absa
226  90 CONTINUE
227  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
228  100 CONTINUE
229  END IF
230  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
231 *
232 * Find normF(A).
233 *
234  scale = zero
235  sum = one
236  IF( k.GT.0 ) THEN
237  IF( lsame( uplo, 'U' ) ) THEN
238  DO 110 j = 2, n
239  CALL classq( min( j-1, k ), ab( max( k+2-j, 1 ), j ),
240  \$ 1, scale, sum )
241  110 CONTINUE
242  l = k + 1
243  ELSE
244  DO 120 j = 1, n - 1
245  CALL classq( min( n-j, k ), ab( 2, j ), 1, scale,
246  \$ sum )
247  120 CONTINUE
248  l = 1
249  END IF
250  sum = 2*sum
251  ELSE
252  l = 1
253  END IF
254  DO 130 j = 1, n
255  IF( real( ab( l, j ) ).NE.zero ) THEN
256  absa = abs( real( ab( l, j ) ) )
257  IF( scale.LT.absa ) THEN
258  sum = one + sum*( scale / absa )**2
259  scale = absa
260  ELSE
261  sum = sum + ( absa / scale )**2
262  END IF
263  END IF
264  130 CONTINUE
265  VALUE = scale*sqrt( sum )
266  END IF
267 *
268  clanhb = VALUE
269  RETURN
270 *
271 * End of CLANHB
272 *
273  END
subroutine classq(n, x, incx, scl, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f90:137
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 clanhb(NORM, UPLO, N, K, AB, LDAB, WORK)
CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhb.f:132