1: *> \brief \b DPBSTF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPBSTF + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbstf.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbstf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KD, LDAB, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION AB( LDAB, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DPBSTF computes a split Cholesky factorization of a real
38: *> symmetric positive definite band matrix A.
39: *>
40: *> This routine is designed to be used in conjunction with DSBGST.
41: *>
42: *> The factorization has the form A = S**T*S where S is a band matrix
43: *> of the same bandwidth as A and the following structure:
44: *>
45: *> S = ( U )
46: *> ( M L )
47: *>
48: *> where U is upper triangular of order m = (n+kd)/2, and L is lower
49: *> triangular of order n-m.
50: *> \endverbatim
51: *
52: * Arguments:
53: * ==========
54: *
55: *> \param[in] UPLO
56: *> \verbatim
57: *> UPLO is CHARACTER*1
58: *> = 'U': Upper triangle of A is stored;
59: *> = 'L': Lower triangle of A is stored.
60: *> \endverbatim
61: *>
62: *> \param[in] N
63: *> \verbatim
64: *> N is INTEGER
65: *> The order of the matrix A. N >= 0.
66: *> \endverbatim
67: *>
68: *> \param[in] KD
69: *> \verbatim
70: *> KD is INTEGER
71: *> The number of superdiagonals of the matrix A if UPLO = 'U',
72: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in,out] AB
76: *> \verbatim
77: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
78: *> On entry, the upper or lower triangle of the symmetric band
79: *> matrix A, stored in the first kd+1 rows of the array. The
80: *> j-th column of A is stored in the j-th column of the array AB
81: *> as follows:
82: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84: *>
85: *> On exit, if INFO = 0, the factor S from the split Cholesky
86: *> factorization A = S**T*S. See Further Details.
87: *> \endverbatim
88: *>
89: *> \param[in] LDAB
90: *> \verbatim
91: *> LDAB is INTEGER
92: *> The leading dimension of the array AB. LDAB >= KD+1.
93: *> \endverbatim
94: *>
95: *> \param[out] INFO
96: *> \verbatim
97: *> INFO is INTEGER
98: *> = 0: successful exit
99: *> < 0: if INFO = -i, the i-th argument had an illegal value
100: *> > 0: if INFO = i, the factorization could not be completed,
101: *> because the updated element a(i,i) was negative; the
102: *> matrix A is not positive definite.
103: *> \endverbatim
104: *
105: * Authors:
106: * ========
107: *
108: *> \author Univ. of Tennessee
109: *> \author Univ. of California Berkeley
110: *> \author Univ. of Colorado Denver
111: *> \author NAG Ltd.
112: *
113: *> \date November 2011
114: *
115: *> \ingroup doubleOTHERcomputational
116: *
117: *> \par Further Details:
118: * =====================
119: *>
120: *> \verbatim
121: *>
122: *> The band storage scheme is illustrated by the following example, when
123: *> N = 7, KD = 2:
124: *>
125: *> S = ( s11 s12 s13 )
126: *> ( s22 s23 s24 )
127: *> ( s33 s34 )
128: *> ( s44 )
129: *> ( s53 s54 s55 )
130: *> ( s64 s65 s66 )
131: *> ( s75 s76 s77 )
132: *>
133: *> If UPLO = 'U', the array AB holds:
134: *>
135: *> on entry: on exit:
136: *>
137: *> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
138: *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
139: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
140: *>
141: *> If UPLO = 'L', the array AB holds:
142: *>
143: *> on entry: on exit:
144: *>
145: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
146: *> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
147: *> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
148: *>
149: *> Array elements marked * are not used by the routine.
150: *> \endverbatim
151: *>
152: * =====================================================================
153: SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
154: *
155: * -- LAPACK computational routine (version 3.4.0) --
156: * -- LAPACK is a software package provided by Univ. of Tennessee, --
157: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
158: * November 2011
159: *
160: * .. Scalar Arguments ..
161: CHARACTER UPLO
162: INTEGER INFO, KD, LDAB, N
163: * ..
164: * .. Array Arguments ..
165: DOUBLE PRECISION AB( LDAB, * )
166: * ..
167: *
168: * =====================================================================
169: *
170: * .. Parameters ..
171: DOUBLE PRECISION ONE, ZERO
172: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
173: * ..
174: * .. Local Scalars ..
175: LOGICAL UPPER
176: INTEGER J, KLD, KM, M
177: DOUBLE PRECISION AJJ
178: * ..
179: * .. External Functions ..
180: LOGICAL LSAME
181: EXTERNAL LSAME
182: * ..
183: * .. External Subroutines ..
184: EXTERNAL DSCAL, DSYR, XERBLA
185: * ..
186: * .. Intrinsic Functions ..
187: INTRINSIC MAX, MIN, SQRT
188: * ..
189: * .. Executable Statements ..
190: *
191: * Test the input parameters.
192: *
193: INFO = 0
194: UPPER = LSAME( UPLO, 'U' )
195: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
196: INFO = -1
197: ELSE IF( N.LT.0 ) THEN
198: INFO = -2
199: ELSE IF( KD.LT.0 ) THEN
200: INFO = -3
201: ELSE IF( LDAB.LT.KD+1 ) THEN
202: INFO = -5
203: END IF
204: IF( INFO.NE.0 ) THEN
205: CALL XERBLA( 'DPBSTF', -INFO )
206: RETURN
207: END IF
208: *
209: * Quick return if possible
210: *
211: IF( N.EQ.0 )
212: $ RETURN
213: *
214: KLD = MAX( 1, LDAB-1 )
215: *
216: * Set the splitting point m.
217: *
218: M = ( N+KD ) / 2
219: *
220: IF( UPPER ) THEN
221: *
222: * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
223: *
224: DO 10 J = N, M + 1, -1
225: *
226: * Compute s(j,j) and test for non-positive-definiteness.
227: *
228: AJJ = AB( KD+1, J )
229: IF( AJJ.LE.ZERO )
230: $ GO TO 50
231: AJJ = SQRT( AJJ )
232: AB( KD+1, J ) = AJJ
233: KM = MIN( J-1, KD )
234: *
235: * Compute elements j-km:j-1 of the j-th column and update the
236: * the leading submatrix within the band.
237: *
238: CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
239: CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
240: $ AB( KD+1, J-KM ), KLD )
241: 10 CONTINUE
242: *
243: * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
244: *
245: DO 20 J = 1, M
246: *
247: * Compute s(j,j) and test for non-positive-definiteness.
248: *
249: AJJ = AB( KD+1, J )
250: IF( AJJ.LE.ZERO )
251: $ GO TO 50
252: AJJ = SQRT( AJJ )
253: AB( KD+1, J ) = AJJ
254: KM = MIN( KD, M-J )
255: *
256: * Compute elements j+1:j+km of the j-th row and update the
257: * trailing submatrix within the band.
258: *
259: IF( KM.GT.0 ) THEN
260: CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
261: CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
262: $ AB( KD+1, J+1 ), KLD )
263: END IF
264: 20 CONTINUE
265: ELSE
266: *
267: * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
268: *
269: DO 30 J = N, M + 1, -1
270: *
271: * Compute s(j,j) and test for non-positive-definiteness.
272: *
273: AJJ = AB( 1, J )
274: IF( AJJ.LE.ZERO )
275: $ GO TO 50
276: AJJ = SQRT( AJJ )
277: AB( 1, J ) = AJJ
278: KM = MIN( J-1, KD )
279: *
280: * Compute elements j-km:j-1 of the j-th row and update the
281: * trailing submatrix within the band.
282: *
283: CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
284: CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
285: $ AB( 1, J-KM ), KLD )
286: 30 CONTINUE
287: *
288: * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
289: *
290: DO 40 J = 1, M
291: *
292: * Compute s(j,j) and test for non-positive-definiteness.
293: *
294: AJJ = AB( 1, J )
295: IF( AJJ.LE.ZERO )
296: $ GO TO 50
297: AJJ = SQRT( AJJ )
298: AB( 1, J ) = AJJ
299: KM = MIN( KD, M-J )
300: *
301: * Compute elements j+1:j+km of the j-th column and update the
302: * trailing submatrix within the band.
303: *
304: IF( KM.GT.0 ) THEN
305: CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
306: CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
307: $ AB( 1, J+1 ), KLD )
308: END IF
309: 40 CONTINUE
310: END IF
311: RETURN
312: *
313: 50 CONTINUE
314: INFO = J
315: RETURN
316: *
317: * End of DPBSTF
318: *
319: END
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