1*> \brief \b DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band 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*       DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
22*                        LDAB, WORK )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          DIAG, NORM, UPLO
26*       INTEGER            K, LDAB, N
27*       ..
28*       .. Array Arguments ..
29*       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> DLANTB  returns the value of the one norm,  or the Frobenius norm, or
39*> the  infinity norm,  or the element of  largest absolute value  of an
40*> n by n triangular band matrix A,  with ( k + 1 ) diagonals.
41*> \endverbatim
42*>
43*> \return DLANTB
44*> \verbatim
45*>
46*>    DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
47*>             (
48*>             ( norm1(A),         NORM = '1', 'O' or 'o'
49*>             (
50*>             ( normI(A),         NORM = 'I' or 'i'
51*>             (
52*>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
53*>
54*> where  norm1  denotes the  one norm of a matrix (maximum column sum),
55*> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
56*> normF  denotes the  Frobenius norm of a matrix (square root of sum of
57*> squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
58*> \endverbatim
59*
60*  Arguments:
61*  ==========
62*
63*> \param[in] NORM
64*> \verbatim
65*>          NORM is CHARACTER*1
66*>          Specifies the value to be returned in DLANTB as described
67*>          above.
68*> \endverbatim
69*>
70*> \param[in] UPLO
71*> \verbatim
72*>          UPLO is CHARACTER*1
73*>          Specifies whether the matrix A is upper or lower triangular.
74*>          = 'U':  Upper triangular
75*>          = 'L':  Lower triangular
76*> \endverbatim
77*>
78*> \param[in] DIAG
79*> \verbatim
80*>          DIAG is CHARACTER*1
81*>          Specifies whether or not the matrix A is unit triangular.
82*>          = 'N':  Non-unit triangular
83*>          = 'U':  Unit triangular
84*> \endverbatim
85*>
86*> \param[in] N
87*> \verbatim
88*>          N is INTEGER
89*>          The order of the matrix A.  N >= 0.  When N = 0, DLANTB is
90*>          set to zero.
91*> \endverbatim
92*>
93*> \param[in] K
94*> \verbatim
95*>          K is INTEGER
96*>          The number of super-diagonals of the matrix A if UPLO = 'U',
97*>          or the number of sub-diagonals of the matrix A if UPLO = 'L'.
98*>          K >= 0.
99*> \endverbatim
100*>
101*> \param[in] AB
102*> \verbatim
103*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
104*>          The upper or lower triangular band matrix A, stored in the
105*>          first k+1 rows of AB.  The j-th column of A is stored
106*>          in the j-th column of the array AB as follows:
107*>          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
108*>          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
109*>          Note that when DIAG = 'U', the elements of the array AB
110*>          corresponding to the diagonal elements of the matrix A are
111*>          not referenced, but are assumed to be one.
112*> \endverbatim
113*>
114*> \param[in] LDAB
115*> \verbatim
116*>          LDAB is INTEGER
117*>          The leading dimension of the array AB.  LDAB >= K+1.
118*> \endverbatim
119*>
120*> \param[out] WORK
121*> \verbatim
122*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
123*>          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
124*>          referenced.
125*> \endverbatim
126*
127*  Authors:
128*  ========
129*
130*> \author Univ. of Tennessee
131*> \author Univ. of California Berkeley
132*> \author Univ. of Colorado Denver
133*> \author NAG Ltd.
134*
135*> \date September 2012
136*
137*> \ingroup doubleOTHERauxiliary
138*
139*  =====================================================================
140      DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB,
141     $                 LDAB, WORK )
142*
143*  -- LAPACK auxiliary routine (version 3.4.2) --
144*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
145*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146*     September 2012
147*
148*     .. Scalar Arguments ..
149      CHARACTER          DIAG, NORM, UPLO
150      INTEGER            K, LDAB, N
151*     ..
152*     .. Array Arguments ..
153      DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
154*     ..
155*
156* =====================================================================
157*
158*     .. Parameters ..
159      DOUBLE PRECISION   ONE, ZERO
160      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
161*     ..
162*     .. Local Scalars ..
163      LOGICAL            UDIAG
164      INTEGER            I, J, L
165      DOUBLE PRECISION   SCALE, SUM, VALUE
166*     ..
167*     .. External Subroutines ..
168      EXTERNAL           DLASSQ
169*     ..
170*     .. External Functions ..
171      LOGICAL            LSAME, DISNAN
172      EXTERNAL           LSAME, DISNAN
173*     ..
174*     .. Intrinsic Functions ..
175      INTRINSIC          ABS, MAX, MIN, SQRT
176*     ..
177*     .. Executable Statements ..
178*
179      IF( N.EQ.0 ) THEN
180         VALUE = ZERO
181      ELSE IF( LSAME( NORM, 'M' ) ) THEN
182*
183*        Find max(abs(A(i,j))).
184*
185         IF( LSAME( DIAG, 'U' ) ) THEN
186            VALUE = ONE
187            IF( LSAME( UPLO, 'U' ) ) THEN
188               DO 20 J = 1, N
189                  DO 10 I = MAX( K+2-J, 1 ), K
190                     SUM = ABS( AB( I, J ) )
191                     IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
192   10             CONTINUE
193   20          CONTINUE
194            ELSE
195               DO 40 J = 1, N
196                  DO 30 I = 2, MIN( N+1-J, K+1 )
197                     SUM = ABS( AB( I, J ) )
198                     IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
199   30             CONTINUE
200   40          CONTINUE
201            END IF
202         ELSE
203            VALUE = ZERO
204            IF( LSAME( UPLO, 'U' ) ) THEN
205               DO 60 J = 1, N
206                  DO 50 I = MAX( K+2-J, 1 ), K + 1
207                     SUM = ABS( AB( I, J ) )
208                     IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
209   50             CONTINUE
210   60          CONTINUE
211            ELSE
212               DO 80 J = 1, N
213                  DO 70 I = 1, MIN( N+1-J, K+1 )
214                     SUM = ABS( AB( I, J ) )
215                     IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
216   70             CONTINUE
217   80          CONTINUE
218            END IF
219         END IF
220      ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN
221*
222*        Find norm1(A).
223*
224         VALUE = ZERO
225         UDIAG = LSAME( DIAG, 'U' )
226         IF( LSAME( UPLO, 'U' ) ) THEN
227            DO 110 J = 1, N
228               IF( UDIAG ) THEN
229                  SUM = ONE
230                  DO 90 I = MAX( K+2-J, 1 ), K
231                     SUM = SUM + ABS( AB( I, J ) )
232   90             CONTINUE
233               ELSE
234                  SUM = ZERO
235                  DO 100 I = MAX( K+2-J, 1 ), K + 1
236                     SUM = SUM + ABS( AB( I, J ) )
237  100             CONTINUE
238               END IF
239               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
240  110       CONTINUE
241         ELSE
242            DO 140 J = 1, N
243               IF( UDIAG ) THEN
244                  SUM = ONE
245                  DO 120 I = 2, MIN( N+1-J, K+1 )
246                     SUM = SUM + ABS( AB( I, J ) )
247  120             CONTINUE
248               ELSE
249                  SUM = ZERO
250                  DO 130 I = 1, MIN( N+1-J, K+1 )
251                     SUM = SUM + ABS( AB( I, J ) )
252  130             CONTINUE
253               END IF
254               IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
255  140       CONTINUE
256         END IF
257      ELSE IF( LSAME( NORM, 'I' ) ) THEN
258*
259*        Find normI(A).
260*
261         VALUE = ZERO
262         IF( LSAME( UPLO, 'U' ) ) THEN
263            IF( LSAME( DIAG, 'U' ) ) THEN
264               DO 150 I = 1, N
265                  WORK( I ) = ONE
266  150          CONTINUE
267               DO 170 J = 1, N
268                  L = K + 1 - J
269                  DO 160 I = MAX( 1, J-K ), J - 1
270                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
271  160             CONTINUE
272  170          CONTINUE
273            ELSE
274               DO 180 I = 1, N
275                  WORK( I ) = ZERO
276  180          CONTINUE
277               DO 200 J = 1, N
278                  L = K + 1 - J
279                  DO 190 I = MAX( 1, J-K ), J
280                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
281  190             CONTINUE
282  200          CONTINUE
283            END IF
284         ELSE
285            IF( LSAME( DIAG, 'U' ) ) THEN
286               DO 210 I = 1, N
287                  WORK( I ) = ONE
288  210          CONTINUE
289               DO 230 J = 1, N
290                  L = 1 - J
291                  DO 220 I = J + 1, MIN( N, J+K )
292                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
293  220             CONTINUE
294  230          CONTINUE
295            ELSE
296               DO 240 I = 1, N
297                  WORK( I ) = ZERO
298  240          CONTINUE
299               DO 260 J = 1, N
300                  L = 1 - J
301                  DO 250 I = J, MIN( N, J+K )
302                     WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) )
303  250             CONTINUE
304  260          CONTINUE
305            END IF
306         END IF
307         DO 270 I = 1, N
308            SUM = WORK( I )
309            IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
310  270    CONTINUE
311      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
312*
313*        Find normF(A).
314*
315         IF( LSAME( UPLO, 'U' ) ) THEN
316            IF( LSAME( DIAG, 'U' ) ) THEN
317               SCALE = ONE
318               SUM = N
319               IF( K.GT.0 ) THEN
320                  DO 280 J = 2, N
321                     CALL DLASSQ( MIN( J-1, K ),
322     $                            AB( MAX( K+2-J, 1 ), J ), 1, SCALE,
323     $                            SUM )
324  280             CONTINUE
325               END IF
326            ELSE
327               SCALE = ZERO
328               SUM = ONE
329               DO 290 J = 1, N
330                  CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ),
331     $                         1, SCALE, SUM )
332  290          CONTINUE
333            END IF
334         ELSE
335            IF( LSAME( DIAG, 'U' ) ) THEN
336               SCALE = ONE
337               SUM = N
338               IF( K.GT.0 ) THEN
339                  DO 300 J = 1, N - 1
340                     CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
341     $                            SUM )
342  300             CONTINUE
343               END IF
344            ELSE
345               SCALE = ZERO
346               SUM = ONE
347               DO 310 J = 1, N
348                  CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE,
349     $                         SUM )
350  310          CONTINUE
351            END IF
352         END IF
353         VALUE = SCALE*SQRT( SUM )
354      END IF
355*
356      DLANTB = VALUE
357      RETURN
358*
359*     End of DLANTB
360*
361      END
362