1: *> \brief \b DLANSP 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 matrix supplied in packed form.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLANSP + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlansp.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlansp.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlansp.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER NORM, UPLO
25: * INTEGER N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION AP( * ), WORK( * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DLANSP returns the value of the one norm, or the Frobenius norm, or
38: *> the infinity norm, or the element of largest absolute value of a
39: *> real symmetric matrix A, supplied in packed form.
40: *> \endverbatim
41: *>
42: *> \return DLANSP
43: *> \verbatim
44: *>
45: *> DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
46: *> (
47: *> ( norm1(A), NORM = '1', 'O' or 'o'
48: *> (
49: *> ( normI(A), NORM = 'I' or 'i'
50: *> (
51: *> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
52: *>
53: *> where norm1 denotes the one norm of a matrix (maximum column sum),
54: *> normI denotes the infinity norm of a matrix (maximum row sum) and
55: *> normF denotes the Frobenius norm of a matrix (square root of sum of
56: *> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
57: *> \endverbatim
58: *
59: * Arguments:
60: * ==========
61: *
62: *> \param[in] NORM
63: *> \verbatim
64: *> NORM is CHARACTER*1
65: *> Specifies the value to be returned in DLANSP as described
66: *> above.
67: *> \endverbatim
68: *>
69: *> \param[in] UPLO
70: *> \verbatim
71: *> UPLO is CHARACTER*1
72: *> Specifies whether the upper or lower triangular part of the
73: *> symmetric matrix A is supplied.
74: *> = 'U': Upper triangular part of A is supplied
75: *> = 'L': Lower triangular part of A is supplied
76: *> \endverbatim
77: *>
78: *> \param[in] N
79: *> \verbatim
80: *> N is INTEGER
81: *> The order of the matrix A. N >= 0. When N = 0, DLANSP is
82: *> set to zero.
83: *> \endverbatim
84: *>
85: *> \param[in] AP
86: *> \verbatim
87: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
88: *> The upper or lower triangle of the symmetric matrix A, packed
89: *> columnwise in a linear array. The j-th column of A is stored
90: *> in the array AP as follows:
91: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
92: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
93: *> \endverbatim
94: *>
95: *> \param[out] WORK
96: *> \verbatim
97: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
98: *> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
99: *> WORK is not referenced.
100: *> \endverbatim
101: *
102: * Authors:
103: * ========
104: *
105: *> \author Univ. of Tennessee
106: *> \author Univ. of California Berkeley
107: *> \author Univ. of Colorado Denver
108: *> \author NAG Ltd.
109: *
110: *> \date December 2016
111: *
112: *> \ingroup doubleOTHERauxiliary
113: *
114: * =====================================================================
115: DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK )
116: *
117: * -- LAPACK auxiliary routine (version 3.7.0) --
118: * -- LAPACK is a software package provided by Univ. of Tennessee, --
119: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120: * December 2016
121: *
122: IMPLICIT NONE
123: * .. Scalar Arguments ..
124: CHARACTER NORM, UPLO
125: INTEGER N
126: * ..
127: * .. Array Arguments ..
128: DOUBLE PRECISION AP( * ), WORK( * )
129: * ..
130: *
131: * =====================================================================
132: *
133: * .. Parameters ..
134: DOUBLE PRECISION ONE, ZERO
135: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
136: * ..
137: * .. Local Scalars ..
138: INTEGER I, J, K
139: DOUBLE PRECISION ABSA, SUM, VALUE
140: * ..
141: * .. Local Arrays ..
142: DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
143: * ..
144: * .. External Functions ..
145: LOGICAL LSAME, DISNAN
146: EXTERNAL LSAME, DISNAN
147: * ..
148: * .. External Subroutines ..
149: EXTERNAL DLASSQ, DCOMBSSQ
150: * ..
151: * .. Intrinsic Functions ..
152: INTRINSIC ABS, SQRT
153: * ..
154: * .. Executable Statements ..
155: *
156: IF( N.EQ.0 ) THEN
157: VALUE = ZERO
158: ELSE IF( LSAME( NORM, 'M' ) ) THEN
159: *
160: * Find max(abs(A(i,j))).
161: *
162: VALUE = ZERO
163: IF( LSAME( UPLO, 'U' ) ) THEN
164: K = 1
165: DO 20 J = 1, N
166: DO 10 I = K, K + J - 1
167: SUM = ABS( AP( I ) )
168: IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
169: 10 CONTINUE
170: K = K + J
171: 20 CONTINUE
172: ELSE
173: K = 1
174: DO 40 J = 1, N
175: DO 30 I = K, K + N - J
176: SUM = ABS( AP( I ) )
177: IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
178: 30 CONTINUE
179: K = K + N - J + 1
180: 40 CONTINUE
181: END IF
182: ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
183: $ ( NORM.EQ.'1' ) ) THEN
184: *
185: * Find normI(A) ( = norm1(A), since A is symmetric).
186: *
187: VALUE = ZERO
188: K = 1
189: IF( LSAME( UPLO, 'U' ) ) THEN
190: DO 60 J = 1, N
191: SUM = ZERO
192: DO 50 I = 1, J - 1
193: ABSA = ABS( AP( K ) )
194: SUM = SUM + ABSA
195: WORK( I ) = WORK( I ) + ABSA
196: K = K + 1
197: 50 CONTINUE
198: WORK( J ) = SUM + ABS( AP( K ) )
199: K = K + 1
200: 60 CONTINUE
201: DO 70 I = 1, N
202: SUM = WORK( I )
203: IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
204: 70 CONTINUE
205: ELSE
206: DO 80 I = 1, N
207: WORK( I ) = ZERO
208: 80 CONTINUE
209: DO 100 J = 1, N
210: SUM = WORK( J ) + ABS( AP( K ) )
211: K = K + 1
212: DO 90 I = J + 1, N
213: ABSA = ABS( AP( K ) )
214: SUM = SUM + ABSA
215: WORK( I ) = WORK( I ) + ABSA
216: K = K + 1
217: 90 CONTINUE
218: IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
219: 100 CONTINUE
220: END IF
221: ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
222: *
223: * Find normF(A).
224: * SSQ(1) is scale
225: * SSQ(2) is sum-of-squares
226: * For better accuracy, sum each column separately.
227: *
228: SSQ( 1 ) = ZERO
229: SSQ( 2 ) = ONE
230: *
231: * Sum off-diagonals
232: *
233: K = 2
234: IF( LSAME( UPLO, 'U' ) ) THEN
235: DO 110 J = 2, N
236: COLSSQ( 1 ) = ZERO
237: COLSSQ( 2 ) = ONE
238: CALL DLASSQ( J-1, AP( K ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
239: CALL DCOMBSSQ( SSQ, COLSSQ )
240: K = K + J
241: 110 CONTINUE
242: ELSE
243: DO 120 J = 1, N - 1
244: COLSSQ( 1 ) = ZERO
245: COLSSQ( 2 ) = ONE
246: CALL DLASSQ( N-J, AP( K ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
247: CALL DCOMBSSQ( SSQ, COLSSQ )
248: K = K + N - J + 1
249: 120 CONTINUE
250: END IF
251: SSQ( 2 ) = 2*SSQ( 2 )
252: *
253: * Sum diagonal
254: *
255: K = 1
256: COLSSQ( 1 ) = ZERO
257: COLSSQ( 2 ) = ONE
258: DO 130 I = 1, N
259: IF( AP( K ).NE.ZERO ) THEN
260: ABSA = ABS( AP( K ) )
261: IF( COLSSQ( 1 ).LT.ABSA ) THEN
262: COLSSQ( 2 ) = ONE + COLSSQ(2)*( COLSSQ(1) / ABSA )**2
263: COLSSQ( 1 ) = ABSA
264: ELSE
265: COLSSQ( 2 ) = COLSSQ( 2 ) + ( ABSA / COLSSQ( 1 ) )**2
266: END IF
267: END IF
268: IF( LSAME( UPLO, 'U' ) ) THEN
269: K = K + I + 1
270: ELSE
271: K = K + N - I + 1
272: END IF
273: 130 CONTINUE
274: CALL DCOMBSSQ( SSQ, COLSSQ )
275: VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
276: END IF
277: *
278: DLANSP = VALUE
279: RETURN
280: *
281: * End of DLANSP
282: *
283: END
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