1: *> \brief \b ZPBSTF
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: * SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KD, LDAB, N
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 AB( LDAB, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZPBSTF computes a split Cholesky factorization of a complex
38: *> Hermitian positive definite band matrix A.
39: *>
40: *> This routine is designed to be used in conjunction with ZHBGST.
41: *>
42: *> The factorization has the form A = S**H*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 COMPLEX*16 array, dimension (LDAB,N)
78: *> On entry, the upper or lower triangle of the Hermitian 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**H*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 complex16OTHERcomputational
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**H s64**H s75**H
138: *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
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**H s23**H s34**H s54 s65 s76 *
147: *> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
148: *>
149: *> Array elements marked * are not used by the routine; s12**H denotes
150: *> conjg(s12); the diagonal elements of S are real.
151: *> \endverbatim
152: *>
153: * =====================================================================
154: SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
155: *
156: * -- LAPACK computational routine (version 3.4.0) --
157: * -- LAPACK is a software package provided by Univ. of Tennessee, --
158: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159: * November 2011
160: *
161: * .. Scalar Arguments ..
162: CHARACTER UPLO
163: INTEGER INFO, KD, LDAB, N
164: * ..
165: * .. Array Arguments ..
166: COMPLEX*16 AB( LDAB, * )
167: * ..
168: *
169: * =====================================================================
170: *
171: * .. Parameters ..
172: DOUBLE PRECISION ONE, ZERO
173: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
174: * ..
175: * .. Local Scalars ..
176: LOGICAL UPPER
177: INTEGER J, KLD, KM, M
178: DOUBLE PRECISION AJJ
179: * ..
180: * .. External Functions ..
181: LOGICAL LSAME
182: EXTERNAL LSAME
183: * ..
184: * .. External Subroutines ..
185: EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV
186: * ..
187: * .. Intrinsic Functions ..
188: INTRINSIC DBLE, MAX, MIN, SQRT
189: * ..
190: * .. Executable Statements ..
191: *
192: * Test the input parameters.
193: *
194: INFO = 0
195: UPPER = LSAME( UPLO, 'U' )
196: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
197: INFO = -1
198: ELSE IF( N.LT.0 ) THEN
199: INFO = -2
200: ELSE IF( KD.LT.0 ) THEN
201: INFO = -3
202: ELSE IF( LDAB.LT.KD+1 ) THEN
203: INFO = -5
204: END IF
205: IF( INFO.NE.0 ) THEN
206: CALL XERBLA( 'ZPBSTF', -INFO )
207: RETURN
208: END IF
209: *
210: * Quick return if possible
211: *
212: IF( N.EQ.0 )
213: $ RETURN
214: *
215: KLD = MAX( 1, LDAB-1 )
216: *
217: * Set the splitting point m.
218: *
219: M = ( N+KD ) / 2
220: *
221: IF( UPPER ) THEN
222: *
223: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
224: *
225: DO 10 J = N, M + 1, -1
226: *
227: * Compute s(j,j) and test for non-positive-definiteness.
228: *
229: AJJ = DBLE( AB( KD+1, J ) )
230: IF( AJJ.LE.ZERO ) THEN
231: AB( KD+1, J ) = AJJ
232: GO TO 50
233: END IF
234: AJJ = SQRT( AJJ )
235: AB( KD+1, J ) = AJJ
236: KM = MIN( J-1, KD )
237: *
238: * Compute elements j-km:j-1 of the j-th column and update the
239: * the leading submatrix within the band.
240: *
241: CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
242: CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
243: $ AB( KD+1, J-KM ), KLD )
244: 10 CONTINUE
245: *
246: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
247: *
248: DO 20 J = 1, M
249: *
250: * Compute s(j,j) and test for non-positive-definiteness.
251: *
252: AJJ = DBLE( AB( KD+1, J ) )
253: IF( AJJ.LE.ZERO ) THEN
254: AB( KD+1, J ) = AJJ
255: GO TO 50
256: END IF
257: AJJ = SQRT( AJJ )
258: AB( KD+1, J ) = AJJ
259: KM = MIN( KD, M-J )
260: *
261: * Compute elements j+1:j+km of the j-th row and update the
262: * trailing submatrix within the band.
263: *
264: IF( KM.GT.0 ) THEN
265: CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
266: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
267: CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
268: $ AB( KD+1, J+1 ), KLD )
269: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
270: END IF
271: 20 CONTINUE
272: ELSE
273: *
274: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
275: *
276: DO 30 J = N, M + 1, -1
277: *
278: * Compute s(j,j) and test for non-positive-definiteness.
279: *
280: AJJ = DBLE( AB( 1, J ) )
281: IF( AJJ.LE.ZERO ) THEN
282: AB( 1, J ) = AJJ
283: GO TO 50
284: END IF
285: AJJ = SQRT( AJJ )
286: AB( 1, J ) = AJJ
287: KM = MIN( J-1, KD )
288: *
289: * Compute elements j-km:j-1 of the j-th row and update the
290: * trailing submatrix within the band.
291: *
292: CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
293: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
294: CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
295: $ AB( 1, J-KM ), KLD )
296: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
297: 30 CONTINUE
298: *
299: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
300: *
301: DO 40 J = 1, M
302: *
303: * Compute s(j,j) and test for non-positive-definiteness.
304: *
305: AJJ = DBLE( AB( 1, J ) )
306: IF( AJJ.LE.ZERO ) THEN
307: AB( 1, J ) = AJJ
308: GO TO 50
309: END IF
310: AJJ = SQRT( AJJ )
311: AB( 1, J ) = AJJ
312: KM = MIN( KD, M-J )
313: *
314: * Compute elements j+1:j+km of the j-th column and update the
315: * trailing submatrix within the band.
316: *
317: IF( KM.GT.0 ) THEN
318: CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
319: CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
320: $ AB( 1, J+1 ), KLD )
321: END IF
322: 40 CONTINUE
323: END IF
324: RETURN
325: *
326: 50 CONTINUE
327: INFO = J
328: RETURN
329: *
330: * End of ZPBSTF
331: *
332: END
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