 LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

## ◆ clanhp()

 real function clanhp ( character NORM, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK )

CLANHP 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 supplied in packed form.

Purpose:
``` CLANHP  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
complex hermitian matrix A,  supplied in packed form.```
Returns
CLANHP
```    CLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A),         NORM = '1', 'O' or 'o'
(
( normI(A),         NORM = 'I' or 'i'
(
( normF(A),         NORM = 'F', 'f', 'E' or 'e'

where  norm1  denotes the  one norm of a matrix (maximum column sum),
normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
normF  denotes the  Frobenius norm of a matrix (square root of sum of
squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.```
Parameters
 [in] NORM ``` NORM is CHARACTER*1 Specifies the value to be returned in CLANHP as described above.``` [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the hermitian matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is supplied``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0. When N = 0, CLANHP is set to zero.``` [in] AP ``` AP is COMPLEX array, dimension (N*(N+1)/2) The upper or lower triangle of the hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero.``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.```
Date
December 2016

Definition at line 119 of file clanhp.f.

119 *
120 * -- LAPACK auxiliary routine (version 3.7.0) --
121 * -- LAPACK is a software package provided by Univ. of Tennessee, --
122 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
123 * December 2016
124 *
125  IMPLICIT NONE
126 * .. Scalar Arguments ..
127  CHARACTER NORM, UPLO
128  INTEGER N
129 * ..
130 * .. Array Arguments ..
131  REAL WORK( * )
132  COMPLEX AP( * )
133 * ..
134 *
135 * =====================================================================
136 *
137 * .. Parameters ..
138  REAL ONE, ZERO
139  parameter( one = 1.0e+0, zero = 0.0e+0 )
140 * ..
141 * .. Local Scalars ..
142  INTEGER I, J, K
143  REAL ABSA, SUM, VALUE
144 * ..
145 * .. Local Arrays ..
146  REAL SSQ( 2 ), COLSSQ( 2 )
147 * ..
148 * .. External Functions ..
149  LOGICAL LSAME, SISNAN
150  EXTERNAL lsame, sisnan
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL classq, scombssq
154 * ..
155 * .. Intrinsic Functions ..
156  INTRINSIC abs, real, sqrt
157 * ..
158 * .. Executable Statements ..
159 *
160  IF( n.EQ.0 ) THEN
161  VALUE = zero
162  ELSE IF( lsame( norm, 'M' ) ) THEN
163 *
164 * Find max(abs(A(i,j))).
165 *
166  VALUE = zero
167  IF( lsame( uplo, 'U' ) ) THEN
168  k = 0
169  DO 20 j = 1, n
170  DO 10 i = k + 1, k + j - 1
171  sum = abs( ap( i ) )
172  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
173  10 CONTINUE
174  k = k + j
175  sum = abs( real( ap( k ) ) )
176  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
177  20 CONTINUE
178  ELSE
179  k = 1
180  DO 40 j = 1, n
181  sum = abs( real( ap( k ) ) )
182  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
183  DO 30 i = k + 1, k + n - j
184  sum = abs( ap( i ) )
185  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
186  30 CONTINUE
187  k = k + n - j + 1
188  40 CONTINUE
189  END IF
190  ELSE IF( ( lsame( norm, 'I' ) ) .OR. ( lsame( norm, 'O' ) ) .OR.
191  \$ ( norm.EQ.'1' ) ) THEN
192 *
193 * Find normI(A) ( = norm1(A), since A is hermitian).
194 *
195  VALUE = zero
196  k = 1
197  IF( lsame( uplo, 'U' ) ) THEN
198  DO 60 j = 1, n
199  sum = zero
200  DO 50 i = 1, j - 1
201  absa = abs( ap( k ) )
202  sum = sum + absa
203  work( i ) = work( i ) + absa
204  k = k + 1
205  50 CONTINUE
206  work( j ) = sum + abs( real( ap( k ) ) )
207  k = k + 1
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( ap( k ) ) )
219  k = k + 1
220  DO 90 i = j + 1, n
221  absa = abs( ap( k ) )
222  sum = sum + absa
223  work( i ) = work( i ) + absa
224  k = k + 1
225  90 CONTINUE
226  IF( VALUE .LT. sum .OR. sisnan( sum ) ) VALUE = sum
227  100 CONTINUE
228  END IF
229  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
230 *
231 * Find normF(A).
232 * SSQ(1) is scale
233 * SSQ(2) is sum-of-squares
234 * For better accuracy, sum each column separately.
235 *
236  ssq( 1 ) = zero
237  ssq( 2 ) = one
238 *
239 * Sum off-diagonals
240 *
241  k = 2
242  IF( lsame( uplo, 'U' ) ) THEN
243  DO 110 j = 2, n
244  colssq( 1 ) = zero
245  colssq( 2 ) = one
246  CALL classq( j-1, ap( k ), 1, colssq( 1 ), colssq( 2 ) )
247  CALL scombssq( ssq, colssq )
248  k = k + j
249  110 CONTINUE
250  ELSE
251  DO 120 j = 1, n - 1
252  colssq( 1 ) = zero
253  colssq( 2 ) = one
254  CALL classq( n-j, ap( k ), 1, colssq( 1 ), colssq( 2 ) )
255  CALL scombssq( ssq, colssq )
256  k = k + n - j + 1
257  120 CONTINUE
258  END IF
259  ssq( 2 ) = 2*ssq( 2 )
260 *
261 * Sum diagonal
262 *
263  k = 1
264  colssq( 1 ) = zero
265  colssq( 2 ) = one
266  DO 130 i = 1, n
267  IF( real( ap( k ) ).NE.zero ) THEN
268  absa = abs( real( ap( k ) ) )
269  IF( colssq( 1 ).LT.absa ) THEN
270  colssq( 2 ) = one + colssq(2)*( colssq(1) / absa )**2
271  colssq( 1 ) = absa
272  ELSE
273  colssq( 2 ) = colssq( 2 ) + ( absa / colssq( 1 ) )**2
274  END IF
275  END IF
276  IF( lsame( uplo, 'U' ) ) THEN
277  k = k + i + 1
278  ELSE
279  k = k + n - i + 1
280  END IF
281  130 CONTINUE
282  CALL scombssq( ssq, colssq )
283  VALUE = ssq( 1 )*sqrt( ssq( 2 ) )
284  END IF
285 *
286  clanhp = VALUE
287  RETURN
288 *
289 * End of CLANHP
290 *
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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
clanhp
real function clanhp(NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhp.f:119
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