LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zlanhb()

 double precision function zlanhb ( character NORM, character UPLO, integer N, integer K, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK )

ZLANHB 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.

Download ZLANHB + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZLANHB  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the element of  largest absolute value  of an
n by n hermitian band matrix A,  with k super-diagonals.```
Returns
ZLANHB
```    ZLANHB = ( 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 ZLANHB as described above.``` [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHB is set to zero.``` [in] K ``` K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.``` [in] AB ``` AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the hermitian band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.```

Definition at line 130 of file zlanhb.f.

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 DOUBLE PRECISION WORK( * )
143 COMPLEX*16 AB( LDAB, * )
144* ..
145*
146* =====================================================================
147*
148* .. Parameters ..
149 DOUBLE PRECISION ONE, ZERO
150 parameter( one = 1.0d+0, zero = 0.0d+0 )
151* ..
152* .. Local Scalars ..
153 INTEGER I, J, L
154 DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
155* ..
156* .. External Functions ..
157 LOGICAL LSAME, DISNAN
158 EXTERNAL lsame, disnan
159* ..
160* .. External Subroutines ..
161 EXTERNAL zlassq
162* ..
163* .. Intrinsic Functions ..
164 INTRINSIC abs, dble, max, min, 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. disnan( sum ) ) VALUE = sum
180 10 CONTINUE
181 sum = abs( dble( ab( k+1, j ) ) )
182 IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
183 20 CONTINUE
184 ELSE
185 DO 40 j = 1, n
186 sum = abs( dble( ab( 1, j ) ) )
187 IF( VALUE .LT. sum .OR. disnan( 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. disnan( 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( dble( ab( k+1, j ) ) )
210 60 CONTINUE
211 DO 70 i = 1, n
212 sum = work( i )
213 IF( VALUE .LT. sum .OR. disnan( 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( dble( 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. disnan( 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 zlassq( 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 zlassq( 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( dble( ab( l, j ) ).NE.zero ) THEN
256 absa = abs( dble( 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 zlanhb = VALUE
269 RETURN
270*
271* End of ZLANHB
272*
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:59
subroutine zlassq(n, x, incx, scl, sumsq)
ZLASSQ updates a sum of squares represented in scaled form.
Definition: zlassq.f90:137
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function zlanhb(NORM, UPLO, N, K, AB, LDAB, WORK)
ZLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhb.f:132
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