1: SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, KD, LDAB, N
11: * ..
12: * .. Array Arguments ..
13: COMPLEX*16 AB( LDAB, * )
14: * ..
15: *
16: * Purpose
17: * =======
18: *
19: * ZPBSTF computes a split Cholesky factorization of a complex
20: * Hermitian positive definite band matrix A.
21: *
22: * This routine is designed to be used in conjunction with ZHBGST.
23: *
24: * The factorization has the form A = S**H*S where S is a band matrix
25: * of the same bandwidth as A and the following structure:
26: *
27: * S = ( U )
28: * ( M L )
29: *
30: * where U is upper triangular of order m = (n+kd)/2, and L is lower
31: * triangular of order n-m.
32: *
33: * Arguments
34: * =========
35: *
36: * UPLO (input) CHARACTER*1
37: * = 'U': Upper triangle of A is stored;
38: * = 'L': Lower triangle of A is stored.
39: *
40: * N (input) INTEGER
41: * The order of the matrix A. N >= 0.
42: *
43: * KD (input) INTEGER
44: * The number of superdiagonals of the matrix A if UPLO = 'U',
45: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
46: *
47: * AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
48: * On entry, the upper or lower triangle of the Hermitian band
49: * matrix A, stored in the first kd+1 rows of the array. The
50: * j-th column of A is stored in the j-th column of the array AB
51: * as follows:
52: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
53: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
54: *
55: * On exit, if INFO = 0, the factor S from the split Cholesky
56: * factorization A = S**H*S. See Further Details.
57: *
58: * LDAB (input) INTEGER
59: * The leading dimension of the array AB. LDAB >= KD+1.
60: *
61: * INFO (output) INTEGER
62: * = 0: successful exit
63: * < 0: if INFO = -i, the i-th argument had an illegal value
64: * > 0: if INFO = i, the factorization could not be completed,
65: * because the updated element a(i,i) was negative; the
66: * matrix A is not positive definite.
67: *
68: * Further Details
69: * ===============
70: *
71: * The band storage scheme is illustrated by the following example, when
72: * N = 7, KD = 2:
73: *
74: * S = ( s11 s12 s13 )
75: * ( s22 s23 s24 )
76: * ( s33 s34 )
77: * ( s44 )
78: * ( s53 s54 s55 )
79: * ( s64 s65 s66 )
80: * ( s75 s76 s77 )
81: *
82: * If UPLO = 'U', the array AB holds:
83: *
84: * on entry: on exit:
85: *
86: * * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
87: * * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
88: * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
89: *
90: * If UPLO = 'L', the array AB holds:
91: *
92: * on entry: on exit:
93: *
94: * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
95: * a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
96: * a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
97: *
98: * Array elements marked * are not used by the routine; s12**H denotes
99: * conjg(s12); the diagonal elements of S are real.
100:
101: *
102: * =====================================================================
103: *
104: * .. Parameters ..
105: DOUBLE PRECISION ONE, ZERO
106: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
107: * ..
108: * .. Local Scalars ..
109: LOGICAL UPPER
110: INTEGER J, KLD, KM, M
111: DOUBLE PRECISION AJJ
112: * ..
113: * .. External Functions ..
114: LOGICAL LSAME
115: EXTERNAL LSAME
116: * ..
117: * .. External Subroutines ..
118: EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV
119: * ..
120: * .. Intrinsic Functions ..
121: INTRINSIC DBLE, MAX, MIN, SQRT
122: * ..
123: * .. Executable Statements ..
124: *
125: * Test the input parameters.
126: *
127: INFO = 0
128: UPPER = LSAME( UPLO, 'U' )
129: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
130: INFO = -1
131: ELSE IF( N.LT.0 ) THEN
132: INFO = -2
133: ELSE IF( KD.LT.0 ) THEN
134: INFO = -3
135: ELSE IF( LDAB.LT.KD+1 ) THEN
136: INFO = -5
137: END IF
138: IF( INFO.NE.0 ) THEN
139: CALL XERBLA( 'ZPBSTF', -INFO )
140: RETURN
141: END IF
142: *
143: * Quick return if possible
144: *
145: IF( N.EQ.0 )
146: $ RETURN
147: *
148: KLD = MAX( 1, LDAB-1 )
149: *
150: * Set the splitting point m.
151: *
152: M = ( N+KD ) / 2
153: *
154: IF( UPPER ) THEN
155: *
156: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
157: *
158: DO 10 J = N, M + 1, -1
159: *
160: * Compute s(j,j) and test for non-positive-definiteness.
161: *
162: AJJ = DBLE( AB( KD+1, J ) )
163: IF( AJJ.LE.ZERO ) THEN
164: AB( KD+1, J ) = AJJ
165: GO TO 50
166: END IF
167: AJJ = SQRT( AJJ )
168: AB( KD+1, J ) = AJJ
169: KM = MIN( J-1, KD )
170: *
171: * Compute elements j-km:j-1 of the j-th column and update the
172: * the leading submatrix within the band.
173: *
174: CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
175: CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
176: $ AB( KD+1, J-KM ), KLD )
177: 10 CONTINUE
178: *
179: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
180: *
181: DO 20 J = 1, M
182: *
183: * Compute s(j,j) and test for non-positive-definiteness.
184: *
185: AJJ = DBLE( AB( KD+1, J ) )
186: IF( AJJ.LE.ZERO ) THEN
187: AB( KD+1, J ) = AJJ
188: GO TO 50
189: END IF
190: AJJ = SQRT( AJJ )
191: AB( KD+1, J ) = AJJ
192: KM = MIN( KD, M-J )
193: *
194: * Compute elements j+1:j+km of the j-th row and update the
195: * trailing submatrix within the band.
196: *
197: IF( KM.GT.0 ) THEN
198: CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
199: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
200: CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
201: $ AB( KD+1, J+1 ), KLD )
202: CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
203: END IF
204: 20 CONTINUE
205: ELSE
206: *
207: * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
208: *
209: DO 30 J = N, M + 1, -1
210: *
211: * Compute s(j,j) and test for non-positive-definiteness.
212: *
213: AJJ = DBLE( AB( 1, J ) )
214: IF( AJJ.LE.ZERO ) THEN
215: AB( 1, J ) = AJJ
216: GO TO 50
217: END IF
218: AJJ = SQRT( AJJ )
219: AB( 1, J ) = AJJ
220: KM = MIN( J-1, KD )
221: *
222: * Compute elements j-km:j-1 of the j-th row and update the
223: * trailing submatrix within the band.
224: *
225: CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
226: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
227: CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
228: $ AB( 1, J-KM ), KLD )
229: CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
230: 30 CONTINUE
231: *
232: * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
233: *
234: DO 40 J = 1, M
235: *
236: * Compute s(j,j) and test for non-positive-definiteness.
237: *
238: AJJ = DBLE( AB( 1, J ) )
239: IF( AJJ.LE.ZERO ) THEN
240: AB( 1, J ) = AJJ
241: GO TO 50
242: END IF
243: AJJ = SQRT( AJJ )
244: AB( 1, J ) = AJJ
245: KM = MIN( KD, M-J )
246: *
247: * Compute elements j+1:j+km of the j-th column and update the
248: * trailing submatrix within the band.
249: *
250: IF( KM.GT.0 ) THEN
251: CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
252: CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
253: $ AB( 1, J+1 ), KLD )
254: END IF
255: 40 CONTINUE
256: END IF
257: RETURN
258: *
259: 50 CONTINUE
260: INFO = J
261: RETURN
262: *
263: * End of ZPBSTF
264: *
265: END
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