1: *> \brief \b ZPFTRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
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7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPFTRF( TRANSR, UPLO, N, A, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER TRANSR, UPLO
25: * INTEGER N, INFO
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 A( 0: * )
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZPFTRF computes the Cholesky factorization of a complex Hermitian
37: *> positive definite matrix A.
38: *>
39: *> The factorization has the form
40: *> A = U**H * U, if UPLO = 'U', or
41: *> A = L * L**H, if UPLO = 'L',
42: *> where U is an upper triangular matrix and L is lower triangular.
43: *>
44: *> This is the block version of the algorithm, calling Level 3 BLAS.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] TRANSR
51: *> \verbatim
52: *> TRANSR is CHARACTER*1
53: *> = 'N': The Normal TRANSR of RFP A is stored;
54: *> = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
55: *> \endverbatim
56: *>
57: *> \param[in] UPLO
58: *> \verbatim
59: *> UPLO is CHARACTER*1
60: *> = 'U': Upper triangle of RFP A is stored;
61: *> = 'L': Lower triangle of RFP A is stored.
62: *> \endverbatim
63: *>
64: *> \param[in] N
65: *> \verbatim
66: *> N is INTEGER
67: *> The order of the matrix A. N >= 0.
68: *> \endverbatim
69: *>
70: *> \param[in,out] A
71: *> \verbatim
72: *> A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
73: *> On entry, the Hermitian matrix A in RFP format. RFP format is
74: *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
75: *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
76: *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
77: *> the Conjugate-transpose of RFP A as defined when
78: *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
79: *> follows: If UPLO = 'U' the RFP A contains the nt elements of
80: *> upper packed A. If UPLO = 'L' the RFP A contains the elements
81: *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
82: *> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
83: *> is odd. See the Note below for more details.
84: *>
85: *> On exit, if INFO = 0, the factor U or L from the Cholesky
86: *> factorization RFP A = U**H*U or RFP A = L*L**H.
87: *> \endverbatim
88: *>
89: *> \param[out] INFO
90: *> \verbatim
91: *> INFO is INTEGER
92: *> = 0: successful exit
93: *> < 0: if INFO = -i, the i-th argument had an illegal value
94: *> > 0: if INFO = i, the leading minor of order i is not
95: *> positive definite, and the factorization could not be
96: *> completed.
97: *>
98: *> Further Notes on RFP Format:
99: *> ============================
100: *>
101: *> We first consider Standard Packed Format when N is even.
102: *> We give an example where N = 6.
103: *>
104: *> AP is Upper AP is Lower
105: *>
106: *> 00 01 02 03 04 05 00
107: *> 11 12 13 14 15 10 11
108: *> 22 23 24 25 20 21 22
109: *> 33 34 35 30 31 32 33
110: *> 44 45 40 41 42 43 44
111: *> 55 50 51 52 53 54 55
112: *>
113: *> Let TRANSR = 'N'. RFP holds AP as follows:
114: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
115: *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
116: *> conjugate-transpose of the first three columns of AP upper.
117: *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
118: *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
119: *> conjugate-transpose of the last three columns of AP lower.
120: *> To denote conjugate we place -- above the element. This covers the
121: *> case N even and TRANSR = 'N'.
122: *>
123: *> RFP A RFP A
124: *>
125: *> -- -- --
126: *> 03 04 05 33 43 53
127: *> -- --
128: *> 13 14 15 00 44 54
129: *> --
130: *> 23 24 25 10 11 55
131: *>
132: *> 33 34 35 20 21 22
133: *> --
134: *> 00 44 45 30 31 32
135: *> -- --
136: *> 01 11 55 40 41 42
137: *> -- -- --
138: *> 02 12 22 50 51 52
139: *>
140: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
141: *> transpose of RFP A above. One therefore gets:
142: *>
143: *> RFP A RFP A
144: *>
145: *> -- -- -- -- -- -- -- -- -- --
146: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
147: *> -- -- -- -- -- -- -- -- -- --
148: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
149: *> -- -- -- -- -- -- -- -- -- --
150: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
151: *>
152: *> We next consider Standard Packed Format when N is odd.
153: *> We give an example where N = 5.
154: *>
155: *> AP is Upper AP is Lower
156: *>
157: *> 00 01 02 03 04 00
158: *> 11 12 13 14 10 11
159: *> 22 23 24 20 21 22
160: *> 33 34 30 31 32 33
161: *> 44 40 41 42 43 44
162: *>
163: *> Let TRANSR = 'N'. RFP holds AP as follows:
164: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
165: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
166: *> conjugate-transpose of the first two columns of AP upper.
167: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
168: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
169: *> conjugate-transpose of the last two columns of AP lower.
170: *> To denote conjugate we place -- above the element. This covers the
171: *> case N odd and TRANSR = 'N'.
172: *>
173: *> RFP A RFP A
174: *>
175: *> -- --
176: *> 02 03 04 00 33 43
177: *> --
178: *> 12 13 14 10 11 44
179: *>
180: *> 22 23 24 20 21 22
181: *> --
182: *> 00 33 34 30 31 32
183: *> -- --
184: *> 01 11 44 40 41 42
185: *>
186: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
187: *> transpose of RFP A above. One therefore gets:
188: *>
189: *> RFP A RFP A
190: *>
191: *> -- -- -- -- -- -- -- -- --
192: *> 02 12 22 00 01 00 10 20 30 40 50
193: *> -- -- -- -- -- -- -- -- --
194: *> 03 13 23 33 11 33 11 21 31 41 51
195: *> -- -- -- -- -- -- -- -- --
196: *> 04 14 24 34 44 43 44 22 32 42 52
197: *> \endverbatim
198: *
199: * Authors:
200: * ========
201: *
202: *> \author Univ. of Tennessee
203: *> \author Univ. of California Berkeley
204: *> \author Univ. of Colorado Denver
205: *> \author NAG Ltd.
206: *
207: *> \ingroup complex16OTHERcomputational
208: *
209: * =====================================================================
210: SUBROUTINE ZPFTRF( TRANSR, UPLO, N, A, INFO )
211: *
212: * -- LAPACK computational routine --
213: * -- LAPACK is a software package provided by Univ. of Tennessee, --
214: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
215: *
216: * .. Scalar Arguments ..
217: CHARACTER TRANSR, UPLO
218: INTEGER N, INFO
219: * ..
220: * .. Array Arguments ..
221: COMPLEX*16 A( 0: * )
222: *
223: * =====================================================================
224: *
225: * .. Parameters ..
226: DOUBLE PRECISION ONE
227: COMPLEX*16 CONE
228: PARAMETER ( ONE = 1.0D+0, CONE = ( 1.0D+0, 0.0D+0 ) )
229: * ..
230: * .. Local Scalars ..
231: LOGICAL LOWER, NISODD, NORMALTRANSR
232: INTEGER N1, N2, K
233: * ..
234: * .. External Functions ..
235: LOGICAL LSAME
236: EXTERNAL LSAME
237: * ..
238: * .. External Subroutines ..
239: EXTERNAL XERBLA, ZHERK, ZPOTRF, ZTRSM
240: * ..
241: * .. Intrinsic Functions ..
242: INTRINSIC MOD
243: * ..
244: * .. Executable Statements ..
245: *
246: * Test the input parameters.
247: *
248: INFO = 0
249: NORMALTRANSR = LSAME( TRANSR, 'N' )
250: LOWER = LSAME( UPLO, 'L' )
251: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
252: INFO = -1
253: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
254: INFO = -2
255: ELSE IF( N.LT.0 ) THEN
256: INFO = -3
257: END IF
258: IF( INFO.NE.0 ) THEN
259: CALL XERBLA( 'ZPFTRF', -INFO )
260: RETURN
261: END IF
262: *
263: * Quick return if possible
264: *
265: IF( N.EQ.0 )
266: $ RETURN
267: *
268: * If N is odd, set NISODD = .TRUE.
269: * If N is even, set K = N/2 and NISODD = .FALSE.
270: *
271: IF( MOD( N, 2 ).EQ.0 ) THEN
272: K = N / 2
273: NISODD = .FALSE.
274: ELSE
275: NISODD = .TRUE.
276: END IF
277: *
278: * Set N1 and N2 depending on LOWER
279: *
280: IF( LOWER ) THEN
281: N2 = N / 2
282: N1 = N - N2
283: ELSE
284: N1 = N / 2
285: N2 = N - N1
286: END IF
287: *
288: * start execution: there are eight cases
289: *
290: IF( NISODD ) THEN
291: *
292: * N is odd
293: *
294: IF( NORMALTRANSR ) THEN
295: *
296: * N is odd and TRANSR = 'N'
297: *
298: IF( LOWER ) THEN
299: *
300: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
301: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
302: * T1 -> a(0), T2 -> a(n), S -> a(n1)
303: *
304: CALL ZPOTRF( 'L', N1, A( 0 ), N, INFO )
305: IF( INFO.GT.0 )
306: $ RETURN
307: CALL ZTRSM( 'R', 'L', 'C', 'N', N2, N1, CONE, A( 0 ), N,
308: $ A( N1 ), N )
309: CALL ZHERK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
310: $ A( N ), N )
311: CALL ZPOTRF( 'U', N2, A( N ), N, INFO )
312: IF( INFO.GT.0 )
313: $ INFO = INFO + N1
314: *
315: ELSE
316: *
317: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
318: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
319: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
320: *
321: CALL ZPOTRF( 'L', N1, A( N2 ), N, INFO )
322: IF( INFO.GT.0 )
323: $ RETURN
324: CALL ZTRSM( 'L', 'L', 'N', 'N', N1, N2, CONE, A( N2 ), N,
325: $ A( 0 ), N )
326: CALL ZHERK( 'U', 'C', N2, N1, -ONE, A( 0 ), N, ONE,
327: $ A( N1 ), N )
328: CALL ZPOTRF( 'U', N2, A( N1 ), N, INFO )
329: IF( INFO.GT.0 )
330: $ INFO = INFO + N1
331: *
332: END IF
333: *
334: ELSE
335: *
336: * N is odd and TRANSR = 'C'
337: *
338: IF( LOWER ) THEN
339: *
340: * SRPA for LOWER, TRANSPOSE and N is odd
341: * T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
342: * T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
343: *
344: CALL ZPOTRF( 'U', N1, A( 0 ), N1, INFO )
345: IF( INFO.GT.0 )
346: $ RETURN
347: CALL ZTRSM( 'L', 'U', 'C', 'N', N1, N2, CONE, A( 0 ), N1,
348: $ A( N1*N1 ), N1 )
349: CALL ZHERK( 'L', 'C', N2, N1, -ONE, A( N1*N1 ), N1, ONE,
350: $ A( 1 ), N1 )
351: CALL ZPOTRF( 'L', N2, A( 1 ), N1, INFO )
352: IF( INFO.GT.0 )
353: $ INFO = INFO + N1
354: *
355: ELSE
356: *
357: * SRPA for UPPER, TRANSPOSE and N is odd
358: * T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
359: * T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
360: *
361: CALL ZPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
362: IF( INFO.GT.0 )
363: $ RETURN
364: CALL ZTRSM( 'R', 'U', 'N', 'N', N2, N1, CONE, A( N2*N2 ),
365: $ N2, A( 0 ), N2 )
366: CALL ZHERK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
367: $ A( N1*N2 ), N2 )
368: CALL ZPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
369: IF( INFO.GT.0 )
370: $ INFO = INFO + N1
371: *
372: END IF
373: *
374: END IF
375: *
376: ELSE
377: *
378: * N is even
379: *
380: IF( NORMALTRANSR ) THEN
381: *
382: * N is even and TRANSR = 'N'
383: *
384: IF( LOWER ) THEN
385: *
386: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
387: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
388: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
389: *
390: CALL ZPOTRF( 'L', K, A( 1 ), N+1, INFO )
391: IF( INFO.GT.0 )
392: $ RETURN
393: CALL ZTRSM( 'R', 'L', 'C', 'N', K, K, CONE, A( 1 ), N+1,
394: $ A( K+1 ), N+1 )
395: CALL ZHERK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
396: $ A( 0 ), N+1 )
397: CALL ZPOTRF( 'U', K, A( 0 ), N+1, INFO )
398: IF( INFO.GT.0 )
399: $ INFO = INFO + K
400: *
401: ELSE
402: *
403: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
404: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
405: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
406: *
407: CALL ZPOTRF( 'L', K, A( K+1 ), N+1, INFO )
408: IF( INFO.GT.0 )
409: $ RETURN
410: CALL ZTRSM( 'L', 'L', 'N', 'N', K, K, CONE, A( K+1 ),
411: $ N+1, A( 0 ), N+1 )
412: CALL ZHERK( 'U', 'C', K, K, -ONE, A( 0 ), N+1, ONE,
413: $ A( K ), N+1 )
414: CALL ZPOTRF( 'U', K, A( K ), N+1, INFO )
415: IF( INFO.GT.0 )
416: $ INFO = INFO + K
417: *
418: END IF
419: *
420: ELSE
421: *
422: * N is even and TRANSR = 'C'
423: *
424: IF( LOWER ) THEN
425: *
426: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
427: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
428: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
429: *
430: CALL ZPOTRF( 'U', K, A( 0+K ), K, INFO )
431: IF( INFO.GT.0 )
432: $ RETURN
433: CALL ZTRSM( 'L', 'U', 'C', 'N', K, K, CONE, A( K ), N1,
434: $ A( K*( K+1 ) ), K )
435: CALL ZHERK( 'L', 'C', K, K, -ONE, A( K*( K+1 ) ), K, ONE,
436: $ A( 0 ), K )
437: CALL ZPOTRF( 'L', K, A( 0 ), K, INFO )
438: IF( INFO.GT.0 )
439: $ INFO = INFO + K
440: *
441: ELSE
442: *
443: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
444: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
445: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
446: *
447: CALL ZPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
448: IF( INFO.GT.0 )
449: $ RETURN
450: CALL ZTRSM( 'R', 'U', 'N', 'N', K, K, CONE,
451: $ A( K*( K+1 ) ), K, A( 0 ), K )
452: CALL ZHERK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
453: $ A( K*K ), K )
454: CALL ZPOTRF( 'L', K, A( K*K ), K, INFO )
455: IF( INFO.GT.0 )
456: $ INFO = INFO + K
457: *
458: END IF
459: *
460: END IF
461: *
462: END IF
463: *
464: RETURN
465: *
466: * End of ZPFTRF
467: *
468: END
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