1: *> \brief \b ZPBSTF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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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: *> \ingroup complex16OTHERcomputational
114: *
115: *> \par Further Details:
116: * =====================
117: *>
118: *> \verbatim
119: *>
120: *> The band storage scheme is illustrated by the following example, when
121: *> N = 7, KD = 2:
122: *>
123: *> S = ( s11 s12 s13 )
124: *> ( s22 s23 s24 )
125: *> ( s33 s34 )
126: *> ( s44 )
127: *> ( s53 s54 s55 )
128: *> ( s64 s65 s66 )
129: *> ( s75 s76 s77 )
130: *>
131: *> If UPLO = 'U', the array AB holds:
132: *>
133: *> on entry: on exit:
134: *>
135: *> * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
136: *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
137: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
138: *>
139: *> If UPLO = 'L', the array AB holds:
140: *>
141: *> on entry: on exit:
142: *>
143: *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
144: *> a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
145: *> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
146: *>
147: *> Array elements marked * are not used by the routine; s12**H denotes
148: *> conjg(s12); the diagonal elements of S are real.
149: *> \endverbatim
150: *>
151: * =====================================================================
152: SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
153: *
154: * -- LAPACK computational routine --
155: * -- LAPACK is a software package provided by Univ. of Tennessee, --
156: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157: *
158: * .. Scalar Arguments ..
159: CHARACTER UPLO
160: INTEGER INFO, KD, LDAB, N
161: * ..
162: * .. Array Arguments ..
163: COMPLEX*16 AB( LDAB, * )
164: * ..
165: *
166: * =====================================================================
167: *
168: * .. Parameters ..
169: DOUBLE PRECISION ONE, ZERO
170: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
171: * ..
172: * .. Local Scalars ..
173: LOGICAL UPPER
174: INTEGER J, KLD, KM, M
175: DOUBLE PRECISION AJJ
176: * ..
177: * .. External Functions ..
178: LOGICAL LSAME
179: EXTERNAL LSAME
180: * ..
181: * .. External Subroutines ..
182: EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV
183: * ..
184: * .. Intrinsic Functions ..
185: INTRINSIC DBLE, MAX, MIN, SQRT
186: * ..
187: * .. Executable Statements ..
188: *
189: * Test the input parameters.
190: *
191: INFO = 0
192: UPPER = LSAME( UPLO, 'U' )
193: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
194: INFO = -1
195: ELSE IF( N.LT.0 ) THEN
196: INFO = -2
197: ELSE IF( KD.LT.0 ) THEN
198: INFO = -3
199: ELSE IF( LDAB.LT.KD+1 ) THEN
200: INFO = -5
201: END IF
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'ZPBSTF', -INFO )
204: RETURN
205: END IF
206: *
207: * Quick return if possible
208: *
209: IF( N.EQ.0 )
210: $ RETURN
211: *
212: KLD = MAX( 1, LDAB-1 )
213: *
214: * Set the splitting point m.
215: *
216: M = ( N+KD ) / 2
217: *
218: IF( UPPER ) THEN
219: *
220: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
221: *
222: DO 10 J = N, M + 1, -1
223: *
224: * Compute s(j,j) and test for non-positive-definiteness.
225: *
226: AJJ = DBLE( AB( KD+1, J ) )
227: IF( AJJ.LE.ZERO ) THEN
228: AB( KD+1, J ) = AJJ
229: GO TO 50
230: END IF
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 ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
239: CALL ZHER( '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**H*U.
244: *
245: DO 20 J = 1, M
246: *
247: * Compute s(j,j) and test for non-positive-definiteness.
248: *
249: AJJ = DBLE( AB( KD+1, J ) )
250: IF( AJJ.LE.ZERO ) THEN
251: AB( KD+1, J ) = AJJ
252: GO TO 50
253: END IF
254: AJJ = SQRT( AJJ )
255: AB( KD+1, J ) = AJJ
256: KM = MIN( KD, M-J )
257: *
258: * Compute elements j+1:j+km of the j-th row and update the
259: * trailing submatrix within the band.
260: *
261: IF( KM.GT.0 ) THEN
262: CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
263: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
264: CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
265: $ AB( KD+1, J+1 ), KLD )
266: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
267: END IF
268: 20 CONTINUE
269: ELSE
270: *
271: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
272: *
273: DO 30 J = N, M + 1, -1
274: *
275: * Compute s(j,j) and test for non-positive-definiteness.
276: *
277: AJJ = DBLE( AB( 1, J ) )
278: IF( AJJ.LE.ZERO ) THEN
279: AB( 1, J ) = AJJ
280: GO TO 50
281: END IF
282: AJJ = SQRT( AJJ )
283: AB( 1, J ) = AJJ
284: KM = MIN( J-1, KD )
285: *
286: * Compute elements j-km:j-1 of the j-th row and update the
287: * trailing submatrix within the band.
288: *
289: CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
290: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
291: CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
292: $ AB( 1, J-KM ), KLD )
293: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
294: 30 CONTINUE
295: *
296: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
297: *
298: DO 40 J = 1, M
299: *
300: * Compute s(j,j) and test for non-positive-definiteness.
301: *
302: AJJ = DBLE( AB( 1, J ) )
303: IF( AJJ.LE.ZERO ) THEN
304: AB( 1, J ) = AJJ
305: GO TO 50
306: END IF
307: AJJ = SQRT( AJJ )
308: AB( 1, J ) = AJJ
309: KM = MIN( KD, M-J )
310: *
311: * Compute elements j+1:j+km of the j-th column and update the
312: * trailing submatrix within the band.
313: *
314: IF( KM.GT.0 ) THEN
315: CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
316: CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
317: $ AB( 1, J+1 ), KLD )
318: END IF
319: 40 CONTINUE
320: END IF
321: RETURN
322: *
323: 50 CONTINUE
324: INFO = J
325: RETURN
326: *
327: * End of ZPBSTF
328: *
329: END
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