LAPACK  3.9.0
LAPACK: Linear Algebra PACKage

◆ zlantr()

double precision function zlantr ( character  NORM,
character  UPLO,
character  DIAG,
integer  M,
integer  N,
complex*16, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( * )  WORK 
)

ZLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

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

Purpose:
 ZLANTR  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 trapezoidal or triangular matrix A.
Returns
ZLANTR 
    ZLANTR = ( 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 ZLANTR as described
          above.
[in]UPLO
          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower trapezoidal.
          = 'U':  Upper trapezoidal
          = 'L':  Lower trapezoidal
          Note that A is triangular instead of trapezoidal if M = N.
[in]DIAG
          DIAG is CHARACTER*1
          Specifies whether or not the matrix A has unit diagonal.
          = 'N':  Non-unit diagonal
          = 'U':  Unit diagonal
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0, and if
          UPLO = 'U', M <= N.  When M = 0, ZLANTR is set to zero.
[in]N
          N is INTEGER
          The number of columns of the matrix A.  N >= 0, and if
          UPLO = 'L', N <= M.  When N = 0, ZLANTR is set to zero.
[in]A
          A is COMPLEX*16 array, dimension (LDA,N)
          The trapezoidal matrix A (A is triangular if M = N).
          If UPLO = 'U', the leading m by n upper trapezoidal part of
          the array A contains the upper trapezoidal matrix, and the
          strictly lower triangular part of A is not referenced.
          If UPLO = 'L', the leading m by n lower trapezoidal part of
          the array A contains the lower trapezoidal matrix, and the
          strictly upper triangular part of A is not referenced.  Note
          that when DIAG = 'U', the diagonal elements of A are not
          referenced and are assumed to be one.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 144 of file zlantr.f.

144 *
145 * -- LAPACK auxiliary routine (version 3.7.0) --
146 * -- LAPACK is a software package provided by Univ. of Tennessee, --
147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148 * December 2016
149 *
150  IMPLICIT NONE
151 * .. Scalar Arguments ..
152  CHARACTER DIAG, NORM, UPLO
153  INTEGER LDA, M, N
154 * ..
155 * .. Array Arguments ..
156  DOUBLE PRECISION WORK( * )
157  COMPLEX*16 A( LDA, * )
158 * ..
159 *
160 * =====================================================================
161 *
162 * .. Parameters ..
163  DOUBLE PRECISION ONE, ZERO
164  parameter( one = 1.0d+0, zero = 0.0d+0 )
165 * ..
166 * .. Local Scalars ..
167  LOGICAL UDIAG
168  INTEGER I, J
169  DOUBLE PRECISION SUM, VALUE
170 * ..
171 * .. Local Arrays ..
172  DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
173 * ..
174 * .. External Functions ..
175  LOGICAL LSAME, DISNAN
176  EXTERNAL lsame, disnan
177 * ..
178 * .. External Subroutines ..
179  EXTERNAL zlassq, dcombssq
180 * ..
181 * .. Intrinsic Functions ..
182  INTRINSIC abs, min, sqrt
183 * ..
184 * .. Executable Statements ..
185 *
186  IF( min( m, n ).EQ.0 ) THEN
187  VALUE = zero
188  ELSE IF( lsame( norm, 'M' ) ) THEN
189 *
190 * Find max(abs(A(i,j))).
191 *
192  IF( lsame( diag, 'U' ) ) THEN
193  VALUE = one
194  IF( lsame( uplo, 'U' ) ) THEN
195  DO 20 j = 1, n
196  DO 10 i = 1, min( m, j-1 )
197  sum = abs( a( i, j ) )
198  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
199  10 CONTINUE
200  20 CONTINUE
201  ELSE
202  DO 40 j = 1, n
203  DO 30 i = j + 1, m
204  sum = abs( a( i, j ) )
205  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
206  30 CONTINUE
207  40 CONTINUE
208  END IF
209  ELSE
210  VALUE = zero
211  IF( lsame( uplo, 'U' ) ) THEN
212  DO 60 j = 1, n
213  DO 50 i = 1, min( m, j )
214  sum = abs( a( i, j ) )
215  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
216  50 CONTINUE
217  60 CONTINUE
218  ELSE
219  DO 80 j = 1, n
220  DO 70 i = j, m
221  sum = abs( a( i, j ) )
222  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
223  70 CONTINUE
224  80 CONTINUE
225  END IF
226  END IF
227  ELSE IF( ( lsame( norm, 'O' ) ) .OR. ( norm.EQ.'1' ) ) THEN
228 *
229 * Find norm1(A).
230 *
231  VALUE = zero
232  udiag = lsame( diag, 'U' )
233  IF( lsame( uplo, 'U' ) ) THEN
234  DO 110 j = 1, n
235  IF( ( udiag ) .AND. ( j.LE.m ) ) THEN
236  sum = one
237  DO 90 i = 1, j - 1
238  sum = sum + abs( a( i, j ) )
239  90 CONTINUE
240  ELSE
241  sum = zero
242  DO 100 i = 1, min( m, j )
243  sum = sum + abs( a( i, j ) )
244  100 CONTINUE
245  END IF
246  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
247  110 CONTINUE
248  ELSE
249  DO 140 j = 1, n
250  IF( udiag ) THEN
251  sum = one
252  DO 120 i = j + 1, m
253  sum = sum + abs( a( i, j ) )
254  120 CONTINUE
255  ELSE
256  sum = zero
257  DO 130 i = j, m
258  sum = sum + abs( a( i, j ) )
259  130 CONTINUE
260  END IF
261  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
262  140 CONTINUE
263  END IF
264  ELSE IF( lsame( norm, 'I' ) ) THEN
265 *
266 * Find normI(A).
267 *
268  IF( lsame( uplo, 'U' ) ) THEN
269  IF( lsame( diag, 'U' ) ) THEN
270  DO 150 i = 1, m
271  work( i ) = one
272  150 CONTINUE
273  DO 170 j = 1, n
274  DO 160 i = 1, min( m, j-1 )
275  work( i ) = work( i ) + abs( a( i, j ) )
276  160 CONTINUE
277  170 CONTINUE
278  ELSE
279  DO 180 i = 1, m
280  work( i ) = zero
281  180 CONTINUE
282  DO 200 j = 1, n
283  DO 190 i = 1, min( m, j )
284  work( i ) = work( i ) + abs( a( i, j ) )
285  190 CONTINUE
286  200 CONTINUE
287  END IF
288  ELSE
289  IF( lsame( diag, 'U' ) ) THEN
290  DO 210 i = 1, n
291  work( i ) = one
292  210 CONTINUE
293  DO 220 i = n + 1, m
294  work( i ) = zero
295  220 CONTINUE
296  DO 240 j = 1, n
297  DO 230 i = j + 1, m
298  work( i ) = work( i ) + abs( a( i, j ) )
299  230 CONTINUE
300  240 CONTINUE
301  ELSE
302  DO 250 i = 1, m
303  work( i ) = zero
304  250 CONTINUE
305  DO 270 j = 1, n
306  DO 260 i = j, m
307  work( i ) = work( i ) + abs( a( i, j ) )
308  260 CONTINUE
309  270 CONTINUE
310  END IF
311  END IF
312  VALUE = zero
313  DO 280 i = 1, m
314  sum = work( i )
315  IF( VALUE .LT. sum .OR. disnan( sum ) ) VALUE = sum
316  280 CONTINUE
317  ELSE IF( ( lsame( norm, 'F' ) ) .OR. ( lsame( norm, 'E' ) ) ) THEN
318 *
319 * Find normF(A).
320 * SSQ(1) is scale
321 * SSQ(2) is sum-of-squares
322 * For better accuracy, sum each column separately.
323 *
324  IF( lsame( uplo, 'U' ) ) THEN
325  IF( lsame( diag, 'U' ) ) THEN
326  ssq( 1 ) = one
327  ssq( 2 ) = min( m, n )
328  DO 290 j = 2, n
329  colssq( 1 ) = zero
330  colssq( 2 ) = one
331  CALL zlassq( min( m, j-1 ), a( 1, j ), 1,
332  $ colssq( 1 ), colssq( 2 ) )
333  CALL dcombssq( ssq, colssq )
334  290 CONTINUE
335  ELSE
336  ssq( 1 ) = zero
337  ssq( 2 ) = one
338  DO 300 j = 1, n
339  colssq( 1 ) = zero
340  colssq( 2 ) = one
341  CALL zlassq( min( m, j ), a( 1, j ), 1,
342  $ colssq( 1 ), colssq( 2 ) )
343  CALL dcombssq( ssq, colssq )
344  300 CONTINUE
345  END IF
346  ELSE
347  IF( lsame( diag, 'U' ) ) THEN
348  ssq( 1 ) = one
349  ssq( 2 ) = min( m, n )
350  DO 310 j = 1, n
351  colssq( 1 ) = zero
352  colssq( 2 ) = one
353  CALL zlassq( m-j, a( min( m, j+1 ), j ), 1,
354  $ colssq( 1 ), colssq( 2 ) )
355  CALL dcombssq( ssq, colssq )
356  310 CONTINUE
357  ELSE
358  ssq( 1 ) = zero
359  ssq( 2 ) = one
360  DO 320 j = 1, n
361  colssq( 1 ) = zero
362  colssq( 2 ) = one
363  CALL zlassq( m-j+1, a( j, j ), 1,
364  $ colssq( 1 ), colssq( 2 ) )
365  CALL dcombssq( ssq, colssq )
366  320 CONTINUE
367  END IF
368  END IF
369  VALUE = ssq( 1 )*sqrt( ssq( 2 ) )
370  END IF
371 *
372  zlantr = VALUE
373  RETURN
374 *
375 * End of ZLANTR
376 *
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zlassq
subroutine zlassq(N, X, INCX, SCALE, SUMSQ)
ZLASSQ updates a sum of squares represented in scaled form.
Definition: zlassq.f:108
disnan
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:61
lsame
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
zlantr
double precision function zlantr(NORM, UPLO, DIAG, M, N, A, LDA, WORK)
ZLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlantr.f:144
dcombssq
subroutine dcombssq(V1, V2)
DCOMBSSQ adds two scaled sum of squares quantities.
Definition: dcombssq.f:62