1*> \brief \b SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download SLANSB + dependencies
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14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slansb.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       REAL             FUNCTION SLANSB( NORM, UPLO, N, K, AB, LDAB,
22*                        WORK )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          NORM, UPLO
26*       INTEGER            K, LDAB, N
27*       ..
28*       .. Array Arguments ..
29*       REAL               AB( LDAB, * ), WORK( * )
30*       ..
31*
32*
33*> \par Purpose:
34*  =============
35*>
36*> \verbatim
37*>
38*> SLANSB  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 symmetric band matrix A,  with k super-diagonals.
41*> \endverbatim
42*>
43*> \return SLANSB
44*> \verbatim
45*>
46*>    SLANSB = ( 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 SLANSB as described
67*>          above.
68*> \endverbatim
69*>
70*> \param[in] UPLO
71*> \verbatim
72*>          UPLO is CHARACTER*1
73*>          Specifies whether the upper or lower triangular part of the
74*>          band matrix A is supplied.
75*>          = 'U':  Upper triangular part is supplied
76*>          = 'L':  Lower triangular part is supplied
77*> \endverbatim
78*>
79*> \param[in] N
80*> \verbatim
81*>          N is INTEGER
82*>          The order of the matrix A.  N >= 0.  When N = 0, SLANSB is
83*>          set to zero.
84*> \endverbatim
85*>
86*> \param[in] K
87*> \verbatim
88*>          K is INTEGER
89*>          The number of super-diagonals or sub-diagonals of the
90*>          band matrix A.  K >= 0.
91*> \endverbatim
92*>
93*> \param[in] AB
94*> \verbatim
95*>          AB is REAL array, dimension (LDAB,N)
96*>          The upper or lower triangle of the symmetric band matrix A,
97*>          stored in the first K+1 rows of AB.  The j-th column of A is
98*>          stored in the j-th column of the array AB as follows:
99*>          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
100*>          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
101*> \endverbatim
102*>
103*> \param[in] LDAB
104*> \verbatim
105*>          LDAB is INTEGER
106*>          The leading dimension of the array AB.  LDAB >= K+1.
107*> \endverbatim
108*>
109*> \param[out] WORK
110*> \verbatim
111*>          WORK is REAL array, dimension (MAX(1,LWORK)),
112*>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
113*>          WORK is not referenced.
114*> \endverbatim
115*
116*  Authors:
117*  ========
118*
119*> \author Univ. of Tennessee
120*> \author Univ. of California Berkeley
121*> \author Univ. of Colorado Denver
122*> \author NAG Ltd.
123*
124*> \ingroup realOTHERauxiliary
125*
126*  =====================================================================
127      REAL             FUNCTION SLANSB( NORM, UPLO, N, K, AB, LDAB,
128     $                 WORK )
129*
130*  -- LAPACK auxiliary routine --
131*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
132*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133*
134      IMPLICIT NONE
135*     .. Scalar Arguments ..
136      CHARACTER          NORM, UPLO
137      INTEGER            K, LDAB, N
138*     ..
139*     .. Array Arguments ..
140      REAL               AB( LDAB, * ), WORK( * )
141*     ..
142*
143* =====================================================================
144*
145*     .. Parameters ..
146      REAL               ONE, ZERO
147      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
148*     ..
149*     .. Local Scalars ..
150      INTEGER            I, J, L
151      REAL               ABSA, SUM, VALUE
152*     ..
153*     .. Local Arrays ..
154      REAL               SSQ( 2 ), COLSSQ( 2 )
155*     ..
156*     .. External Functions ..
157      LOGICAL            LSAME, SISNAN
158      EXTERNAL           LSAME, SISNAN
159*     ..
160*     .. External Subroutines ..
161      EXTERNAL           SLASSQ, SCOMBSSQ
162*     ..
163*     .. Intrinsic Functions ..
164      INTRINSIC          ABS, 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 + 1
178                  SUM = ABS( AB( I, J ) )
179                  IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
180   10          CONTINUE
181   20       CONTINUE
182         ELSE
183            DO 40 J = 1, N
184               DO 30 I = 1, MIN( N+1-J, K+1 )
185                  SUM = ABS( AB( I, J ) )
186                  IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
187   30          CONTINUE
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 symmetric).
194*
195         VALUE = ZERO
196         IF( LSAME( UPLO, 'U' ) ) THEN
197            DO 60 J = 1, N
198               SUM = ZERO
199               L = K + 1 - J
200               DO 50 I = MAX( 1, J-K ), J - 1
201                  ABSA = ABS( AB( L+I, J ) )
202                  SUM = SUM + ABSA
203                  WORK( I ) = WORK( I ) + ABSA
204   50          CONTINUE
205               WORK( J ) = SUM + ABS( AB( K+1, J ) )
206   60       CONTINUE
207            DO 70 I = 1, N
208               SUM = WORK( I )
209               IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
210   70       CONTINUE
211         ELSE
212            DO 80 I = 1, N
213               WORK( I ) = ZERO
214   80       CONTINUE
215            DO 100 J = 1, N
216               SUM = WORK( J ) + ABS( AB( 1, J ) )
217               L = 1 - J
218               DO 90 I = J + 1, MIN( N, J+K )
219                  ABSA = ABS( AB( L+I, J ) )
220                  SUM = SUM + ABSA
221                  WORK( I ) = WORK( I ) + ABSA
222   90          CONTINUE
223               IF( VALUE .LT. SUM .OR. SISNAN( SUM ) ) VALUE = SUM
224  100       CONTINUE
225         END IF
226      ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
227*
228*        Find normF(A).
229*        SSQ(1) is scale
230*        SSQ(2) is sum-of-squares
231*        For better accuracy, sum each column separately.
232*
233         SSQ( 1 ) = ZERO
234         SSQ( 2 ) = ONE
235*
236*        Sum off-diagonals
237*
238         IF( K.GT.0 ) THEN
239            IF( LSAME( UPLO, 'U' ) ) THEN
240               DO 110 J = 2, N
241                  COLSSQ( 1 ) = ZERO
242                  COLSSQ( 2 ) = ONE
243                  CALL SLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
244     $                         1, COLSSQ( 1 ), COLSSQ( 2 ) )
245                  CALL SCOMBSSQ( SSQ, COLSSQ )
246  110          CONTINUE
247               L = K + 1
248            ELSE
249               DO 120 J = 1, N - 1
250                  COLSSQ( 1 ) = ZERO
251                  COLSSQ( 2 ) = ONE
252                  CALL SLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
253     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
254                  CALL SCOMBSSQ( SSQ, COLSSQ )
255  120          CONTINUE
256               L = 1
257            END IF
258            SSQ( 2 ) = 2*SSQ( 2 )
259         ELSE
260            L = 1
261         END IF
262*
263*        Sum diagonal
264*
265         COLSSQ( 1 ) = ZERO
266         COLSSQ( 2 ) = ONE
267         CALL SLASSQ( N, AB( L, 1 ), LDAB, COLSSQ( 1 ), COLSSQ( 2 ) )
268         CALL SCOMBSSQ( SSQ, COLSSQ )
269         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
270      END IF
271*
272      SLANSB = VALUE
273      RETURN
274*
275*     End of SLANSB
276*
277      END
278