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
9: *> Download DLANTB + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlantb.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlantb.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlantb.f">
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 December 2016
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 3.7.0) --
144: * -- LAPACK is a software package provided by Univ. of Tennessee, --
145: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146: * December 2016
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
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