1*> \brief \b CLANTR 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.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       REAL             FUNCTION CLANTR( NORM, UPLO, DIAG, M, N, A, LDA,
22*                        WORK )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          DIAG, NORM, UPLO
26*       INTEGER            LDA, M, N
27*       ..
28*       .. Array Arguments ..
29*       REAL               WORK( * )
30*       COMPLEX            A( LDA, * )
31*       ..
32*
33*
34*> \par Purpose:
35*  =============
36*>
37*> \verbatim
38*>
39*> CLANTR  returns the value of the one norm,  or the Frobenius norm, or
40*> the  infinity norm,  or the  element of  largest absolute value  of a
41*> trapezoidal or triangular matrix A.
42*> \endverbatim
43*>
44*> \return CLANTR
45*> \verbatim
46*>
47*>    CLANTR = ( 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 CLANTR as described
68*>          above.
69*> \endverbatim
70*>
71*> \param[in] UPLO
72*> \verbatim
73*>          UPLO is CHARACTER*1
74*>          Specifies whether the matrix A is upper or lower trapezoidal.
75*>          = 'U':  Upper trapezoidal
76*>          = 'L':  Lower trapezoidal
77*>          Note that A is triangular instead of trapezoidal if M = N.
78*> \endverbatim
79*>
80*> \param[in] DIAG
81*> \verbatim
82*>          DIAG is CHARACTER*1
83*>          Specifies whether or not the matrix A has unit diagonal.
84*>          = 'N':  Non-unit diagonal
85*>          = 'U':  Unit diagonal
86*> \endverbatim
87*>
88*> \param[in] M
89*> \verbatim
90*>          M is INTEGER
91*>          The number of rows of the matrix A.  M >= 0, and if
92*>          UPLO = 'U', M <= N.  When M = 0, CLANTR is set to zero.
93*> \endverbatim
94*>
95*> \param[in] N
96*> \verbatim
97*>          N is INTEGER
98*>          The number of columns of the matrix A.  N >= 0, and if
99*>          UPLO = 'L', N <= M.  When N = 0, CLANTR is set to zero.
100*> \endverbatim
101*>
102*> \param[in] A
103*> \verbatim
104*>          A is COMPLEX array, dimension (LDA,N)
105*>          The trapezoidal matrix A (A is triangular if M = N).
106*>          If UPLO = 'U', the leading m by n upper trapezoidal part of
107*>          the array A contains the upper trapezoidal matrix, and the
108*>          strictly lower triangular part of A is not referenced.
109*>          If UPLO = 'L', the leading m by n lower trapezoidal part of
110*>          the array A contains the lower trapezoidal matrix, and the
111*>          strictly upper triangular part of A is not referenced.  Note
112*>          that when DIAG = 'U', the diagonal elements of A are not
113*>          referenced and are assumed to be one.
114*> \endverbatim
115*>
116*> \param[in] LDA
117*> \verbatim
118*>          LDA is INTEGER
119*>          The leading dimension of the array A.  LDA >= max(M,1).
120*> \endverbatim
121*>
122*> \param[out] WORK
123*> \verbatim
124*>          WORK is REAL array, dimension (MAX(1,LWORK)),
125*>          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
126*>          referenced.
127*> \endverbatim
128*
129*  Authors:
130*  ========
131*
132*> \author Univ. of Tennessee
133*> \author Univ. of California Berkeley
134*> \author Univ. of Colorado Denver
135*> \author NAG Ltd.
136*
137*> \ingroup complexOTHERauxiliary
138*
139*  =====================================================================
140      REAL             FUNCTION CLANTR( NORM, UPLO, DIAG, M, N, A, LDA,
141     $                 WORK )
142*
143*  -- LAPACK auxiliary routine --
144*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
145*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146*
147      IMPLICIT NONE
148*     .. Scalar Arguments ..
149      CHARACTER          DIAG, NORM, UPLO
150      INTEGER            LDA, M, N
151*     ..
152*     .. Array Arguments ..
153      REAL               WORK( * )
154      COMPLEX            A( LDA, * )
155*     ..
156*
157* =====================================================================
158*
159*     .. Parameters ..
160      REAL               ONE, ZERO
161      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
162*     ..
163*     .. Local Scalars ..
164      LOGICAL            UDIAG
165      INTEGER            I, J
166      REAL               SUM, VALUE
167*     ..
168*     .. Local Arrays ..
169      REAL               SSQ( 2 ), COLSSQ( 2 )
170*     ..
171*     .. External Functions ..
172      LOGICAL            LSAME, SISNAN
173      EXTERNAL           LSAME, SISNAN
174*     ..
175*     .. External Subroutines ..
176      EXTERNAL           CLASSQ, SCOMBSSQ
177*     ..
178*     .. Intrinsic Functions ..
179      INTRINSIC          ABS, MIN, SQRT
180*     ..
181*     .. Executable Statements ..
182*
183      IF( MIN( M, N ).EQ.0 ) THEN
184         VALUE = ZERO
185      ELSE IF( LSAME( NORM, 'M' ) ) THEN
186*
187*        Find max(abs(A(i,j))).
188*
189         IF( LSAME( DIAG, 'U' ) ) THEN
190            VALUE = ONE
191            IF( LSAME( UPLO, 'U' ) ) THEN
192               DO 20 J = 1, N
193                  DO 10 I = 1, MIN( M, J-1 )
194                     SUM = ABS( A( I, J ) )
195                     IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
196   10             CONTINUE
197   20          CONTINUE
198            ELSE
199               DO 40 J = 1, N
200                  DO 30 I = J + 1, M
201                     SUM = ABS( A( I, J ) )
202                     IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
203   30             CONTINUE
204   40          CONTINUE
205            END IF
206         ELSE
207            VALUE = ZERO
208            IF( LSAME( UPLO, 'U' ) ) THEN
209               DO 60 J = 1, N
210                  DO 50 I = 1, MIN( M, J )
211                     SUM = ABS( A( I, J ) )
212                     IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
213   50             CONTINUE
214   60          CONTINUE
215            ELSE
216               DO 80 J = 1, N
217                  DO 70 I = J, M
218                     SUM = ABS( A( I, J ) )
219                     IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
220   70             CONTINUE
221   80          CONTINUE
222            END IF
223         END IF
224      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
225*
226*        Find norm1(A).
227*
228         VALUE = ZERO
229         UDIAG = LSAME( DIAG, 'U' )
230         IF( LSAME( UPLO, 'U' ) ) THEN
231            DO 110 J = 1, N
232               IF( ( UDIAG ) .AND. ( J.LE.M ) ) THEN
233                  SUM = ONE
234                  DO 90 I = 1, J - 1
235                     SUM = SUM + ABS( A( I, J ) )
236   90             CONTINUE
237               ELSE
238                  SUM = ZERO
239                  DO 100 I = 1, MIN( M, J )
240                     SUM = SUM + ABS( A( I, J ) )
241  100             CONTINUE
242               END IF
243               IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
244  110       CONTINUE
245         ELSE
246            DO 140 J = 1, N
247               IF( UDIAG ) THEN
248                  SUM = ONE
249                  DO 120 I = J + 1, M
250                     SUM = SUM + ABS( A( I, J ) )
251  120             CONTINUE
252               ELSE
253                  SUM = ZERO
254                  DO 130 I = J, M
255                     SUM = SUM + ABS( A( I, J ) )
256  130             CONTINUE
257               END IF
258               IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
259  140       CONTINUE
260         END IF
261      ELSE IF( LSAME( NORM, 'I' ) ) THEN
262*
263*        Find normI(A).
264*
265         IF( LSAME( UPLO, 'U' ) ) THEN
266            IF( LSAME( DIAG, 'U' ) ) THEN
267               DO 150 I = 1, M
268                  WORK( I ) = ONE
269  150          CONTINUE
270               DO 170 J = 1, N
271                  DO 160 I = 1, MIN( M, J-1 )
272                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
273  160             CONTINUE
274  170          CONTINUE
275            ELSE
276               DO 180 I = 1, M
277                  WORK( I ) = ZERO
278  180          CONTINUE
279               DO 200 J = 1, N
280                  DO 190 I = 1, MIN( M, J )
281                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
282  190             CONTINUE
283  200          CONTINUE
284            END IF
285         ELSE
286            IF( LSAME( DIAG, 'U' ) ) THEN
287               DO 210 I = 1, MIN( M, N )
288                  WORK( I ) = ONE
289  210          CONTINUE
290               DO 220 I = N + 1, M
291                  WORK( I ) = ZERO
292  220          CONTINUE
293               DO 240 J = 1, N
294                  DO 230 I = J + 1, M
295                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
296  230             CONTINUE
297  240          CONTINUE
298            ELSE
299               DO 250 I = 1, M
300                  WORK( I ) = ZERO
301  250          CONTINUE
302               DO 270 J = 1, N
303                  DO 260 I = J, M
304                     WORK( I ) = WORK( I ) + ABS( A( I, J ) )
305  260             CONTINUE
306  270          CONTINUE
307            END IF
308         END IF
309         VALUE = ZERO
310         DO 280 I = 1, M
311            SUM = WORK( I )
312            IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
313  280    CONTINUE
314      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
315*
316*        Find normF(A).
317*        SSQ(1) is scale
318*        SSQ(2) is sum-of-squares
319*        For better accuracy, sum each column separately.
320*
321         IF( LSAME( UPLO, 'U' ) ) THEN
322            IF( LSAME( DIAG, 'U' ) ) THEN
323               SSQ( 1 ) = ONE
324               SSQ( 2 ) = MIN( M, N )
325               DO 290 J = 2, N
326                  COLSSQ( 1 ) = ZERO
327                  COLSSQ( 2 ) = ONE
328                  CALL CLASSQ( MIN( M, J-1 ), A( 1, J ), 1,
329     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
330                  CALL SCOMBSSQ( SSQ, COLSSQ )
331  290          CONTINUE
332            ELSE
333               SSQ( 1 ) = ZERO
334               SSQ( 2 ) = ONE
335               DO 300 J = 1, N
336                  COLSSQ( 1 ) = ZERO
337                  COLSSQ( 2 ) = ONE
338                  CALL CLASSQ( MIN( M, J ), A( 1, J ), 1,
339     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
340                  CALL SCOMBSSQ( SSQ, COLSSQ )
341  300          CONTINUE
342            END IF
343         ELSE
344            IF( LSAME( DIAG, 'U' ) ) THEN
345               SSQ( 1 ) = ONE
346               SSQ( 2 ) = MIN( M, N )
347               DO 310 J = 1, N
348                  COLSSQ( 1 ) = ZERO
349                  COLSSQ( 2 ) = ONE
350                  CALL CLASSQ( M-J, A( MIN( M, J+1 ), J ), 1,
351     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
352                  CALL SCOMBSSQ( SSQ, COLSSQ )
353  310          CONTINUE
354            ELSE
355               SSQ( 1 ) = ZERO
356               SSQ( 2 ) = ONE
357               DO 320 J = 1, N
358                  COLSSQ( 1 ) = ZERO
359                  COLSSQ( 2 ) = ONE
360                  CALL CLASSQ( M-J+1, A( J, J ), 1,
361     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
362                  CALL SCOMBSSQ( SSQ, COLSSQ )
363  320          CONTINUE
364            END IF
365         END IF
366         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
367      END IF
368*
369      CLANTR = VALUE
370      RETURN
371*
372*     End of CLANTR
373*
374      END
375