1: SUBROUTINE ZPFTRF( TRANSR, UPLO, N, A, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: *
5: * -- Contributed by Fred Gustavson of the IBM Watson Research Center --
6: * -- November 2008 --
7: *
8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
9: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
10: *
11: * ..
12: * .. Scalar Arguments ..
13: CHARACTER TRANSR, UPLO
14: INTEGER N, INFO
15: * ..
16: * .. Array Arguments ..
17: COMPLEX*16 A( 0: * )
18: *
19: * Purpose
20: * =======
21: *
22: * ZPFTRF computes the Cholesky factorization of a complex Hermitian
23: * positive definite matrix A.
24: *
25: * The factorization has the form
26: * A = U**H * U, if UPLO = 'U', or
27: * A = L * L**H, if UPLO = 'L',
28: * where U is an upper triangular matrix and L is lower triangular.
29: *
30: * This is the block version of the algorithm, calling Level 3 BLAS.
31: *
32: * Arguments
33: * =========
34: *
35: * TRANSR (input) CHARACTER
36: * = 'N': The Normal TRANSR of RFP A is stored;
37: * = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
38: *
39: * UPLO (input) CHARACTER
40: * = 'U': Upper triangle of RFP A is stored;
41: * = 'L': Lower triangle of RFP A is stored.
42: *
43: * N (input) INTEGER
44: * The order of the matrix A. N >= 0.
45: *
46: * A (input/output) COMPLEX array, dimension ( N*(N+1)/2 );
47: * On entry, the Hermitian matrix A in RFP format. RFP format is
48: * described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
49: * then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
50: * (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
51: * the Conjugate-transpose of RFP A as defined when
52: * TRANSR = 'N'. The contents of RFP A are defined by UPLO as
53: * follows: If UPLO = 'U' the RFP A contains the nt elements of
54: * upper packed A. If UPLO = 'L' the RFP A contains the elements
55: * of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
56: * 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
57: * is odd. See the Note below for more details.
58: *
59: * On exit, if INFO = 0, the factor U or L from the Cholesky
60: * factorization RFP A = U**H*U or RFP A = L*L**H.
61: *
62: * INFO (output) INTEGER
63: * = 0: successful exit
64: * < 0: if INFO = -i, the i-th argument had an illegal value
65: * > 0: if INFO = i, the leading minor of order i is not
66: * positive definite, and the factorization could not be
67: * completed.
68: *
69: * Further Notes on RFP Format:
70: * ============================
71: *
72: * We first consider Standard Packed Format when N is even.
73: * We give an example where N = 6.
74: *
75: * AP is Upper AP is Lower
76: *
77: * 00 01 02 03 04 05 00
78: * 11 12 13 14 15 10 11
79: * 22 23 24 25 20 21 22
80: * 33 34 35 30 31 32 33
81: * 44 45 40 41 42 43 44
82: * 55 50 51 52 53 54 55
83: *
84: *
85: * Let TRANSR = 'N'. RFP holds AP as follows:
86: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
87: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
88: * conjugate-transpose of the first three columns of AP upper.
89: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
90: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
91: * conjugate-transpose of the last three columns of AP lower.
92: * To denote conjugate we place -- above the element. This covers the
93: * case N even and TRANSR = 'N'.
94: *
95: * RFP A RFP A
96: *
97: * -- -- --
98: * 03 04 05 33 43 53
99: * -- --
100: * 13 14 15 00 44 54
101: * --
102: * 23 24 25 10 11 55
103: *
104: * 33 34 35 20 21 22
105: * --
106: * 00 44 45 30 31 32
107: * -- --
108: * 01 11 55 40 41 42
109: * -- -- --
110: * 02 12 22 50 51 52
111: *
112: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
113: * transpose of RFP A above. One therefore gets:
114: *
115: *
116: * RFP A RFP A
117: *
118: * -- -- -- -- -- -- -- -- -- --
119: * 03 13 23 33 00 01 02 33 00 10 20 30 40 50
120: * -- -- -- -- -- -- -- -- -- --
121: * 04 14 24 34 44 11 12 43 44 11 21 31 41 51
122: * -- -- -- -- -- -- -- -- -- --
123: * 05 15 25 35 45 55 22 53 54 55 22 32 42 52
124: *
125: *
126: * We next consider Standard Packed Format when N is odd.
127: * We give an example where N = 5.
128: *
129: * AP is Upper AP is Lower
130: *
131: * 00 01 02 03 04 00
132: * 11 12 13 14 10 11
133: * 22 23 24 20 21 22
134: * 33 34 30 31 32 33
135: * 44 40 41 42 43 44
136: *
137: *
138: * Let TRANSR = 'N'. RFP holds AP as follows:
139: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
140: * three columns of AP upper. The lower triangle A(3:4,0:1) consists of
141: * conjugate-transpose of the first two columns of AP upper.
142: * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
143: * three columns of AP lower. The upper triangle A(0:1,1:2) consists of
144: * conjugate-transpose of the last two columns of AP lower.
145: * To denote conjugate we place -- above the element. This covers the
146: * case N odd and TRANSR = 'N'.
147: *
148: * RFP A RFP A
149: *
150: * -- --
151: * 02 03 04 00 33 43
152: * --
153: * 12 13 14 10 11 44
154: *
155: * 22 23 24 20 21 22
156: * --
157: * 00 33 34 30 31 32
158: * -- --
159: * 01 11 44 40 41 42
160: *
161: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
162: * transpose of RFP A above. One therefore gets:
163: *
164: *
165: * RFP A RFP A
166: *
167: * -- -- -- -- -- -- -- -- --
168: * 02 12 22 00 01 00 10 20 30 40 50
169: * -- -- -- -- -- -- -- -- --
170: * 03 13 23 33 11 33 11 21 31 41 51
171: * -- -- -- -- -- -- -- -- --
172: * 04 14 24 34 44 43 44 22 32 42 52
173: *
174: * =====================================================================
175: *
176: * .. Parameters ..
177: DOUBLE PRECISION ONE
178: COMPLEX*16 CONE
179: PARAMETER ( ONE = 1.0D+0, CONE = ( 1.0D+0, 0.0D+0 ) )
180: * ..
181: * .. Local Scalars ..
182: LOGICAL LOWER, NISODD, NORMALTRANSR
183: INTEGER N1, N2, K
184: * ..
185: * .. External Functions ..
186: LOGICAL LSAME
187: EXTERNAL LSAME
188: * ..
189: * .. External Subroutines ..
190: EXTERNAL XERBLA, ZHERK, ZPOTRF, ZTRSM
191: * ..
192: * .. Intrinsic Functions ..
193: INTRINSIC MOD
194: * ..
195: * .. Executable Statements ..
196: *
197: * Test the input parameters.
198: *
199: INFO = 0
200: NORMALTRANSR = LSAME( TRANSR, 'N' )
201: LOWER = LSAME( UPLO, 'L' )
202: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
203: INFO = -1
204: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
205: INFO = -2
206: ELSE IF( N.LT.0 ) THEN
207: INFO = -3
208: END IF
209: IF( INFO.NE.0 ) THEN
210: CALL XERBLA( 'ZPFTRF', -INFO )
211: RETURN
212: END IF
213: *
214: * Quick return if possible
215: *
216: IF( N.EQ.0 )
217: + RETURN
218: *
219: * If N is odd, set NISODD = .TRUE.
220: * If N is even, set K = N/2 and NISODD = .FALSE.
221: *
222: IF( MOD( N, 2 ).EQ.0 ) THEN
223: K = N / 2
224: NISODD = .FALSE.
225: ELSE
226: NISODD = .TRUE.
227: END IF
228: *
229: * Set N1 and N2 depending on LOWER
230: *
231: IF( LOWER ) THEN
232: N2 = N / 2
233: N1 = N - N2
234: ELSE
235: N1 = N / 2
236: N2 = N - N1
237: END IF
238: *
239: * start execution: there are eight cases
240: *
241: IF( NISODD ) THEN
242: *
243: * N is odd
244: *
245: IF( NORMALTRANSR ) THEN
246: *
247: * N is odd and TRANSR = 'N'
248: *
249: IF( LOWER ) THEN
250: *
251: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
252: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
253: * T1 -> a(0), T2 -> a(n), S -> a(n1)
254: *
255: CALL ZPOTRF( 'L', N1, A( 0 ), N, INFO )
256: IF( INFO.GT.0 )
257: + RETURN
258: CALL ZTRSM( 'R', 'L', 'C', 'N', N2, N1, CONE, A( 0 ), N,
259: + A( N1 ), N )
260: CALL ZHERK( 'U', 'N', N2, N1, -ONE, A( N1 ), N, ONE,
261: + A( N ), N )
262: CALL ZPOTRF( 'U', N2, A( N ), N, INFO )
263: IF( INFO.GT.0 )
264: + INFO = INFO + N1
265: *
266: ELSE
267: *
268: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
269: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
270: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
271: *
272: CALL ZPOTRF( 'L', N1, A( N2 ), N, INFO )
273: IF( INFO.GT.0 )
274: + RETURN
275: CALL ZTRSM( 'L', 'L', 'N', 'N', N1, N2, CONE, A( N2 ), N,
276: + A( 0 ), N )
277: CALL ZHERK( 'U', 'C', N2, N1, -ONE, A( 0 ), N, ONE,
278: + A( N1 ), N )
279: CALL ZPOTRF( 'U', N2, A( N1 ), N, INFO )
280: IF( INFO.GT.0 )
281: + INFO = INFO + N1
282: *
283: END IF
284: *
285: ELSE
286: *
287: * N is odd and TRANSR = 'C'
288: *
289: IF( LOWER ) THEN
290: *
291: * SRPA for LOWER, TRANSPOSE and N is odd
292: * T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
293: * T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
294: *
295: CALL ZPOTRF( 'U', N1, A( 0 ), N1, INFO )
296: IF( INFO.GT.0 )
297: + RETURN
298: CALL ZTRSM( 'L', 'U', 'C', 'N', N1, N2, CONE, A( 0 ), N1,
299: + A( N1*N1 ), N1 )
300: CALL ZHERK( 'L', 'C', N2, N1, -ONE, A( N1*N1 ), N1, ONE,
301: + A( 1 ), N1 )
302: CALL ZPOTRF( 'L', N2, A( 1 ), N1, INFO )
303: IF( INFO.GT.0 )
304: + INFO = INFO + N1
305: *
306: ELSE
307: *
308: * SRPA for UPPER, TRANSPOSE and N is odd
309: * T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
310: * T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
311: *
312: CALL ZPOTRF( 'U', N1, A( N2*N2 ), N2, INFO )
313: IF( INFO.GT.0 )
314: + RETURN
315: CALL ZTRSM( 'R', 'U', 'N', 'N', N2, N1, CONE, A( N2*N2 ),
316: + N2, A( 0 ), N2 )
317: CALL ZHERK( 'L', 'N', N2, N1, -ONE, A( 0 ), N2, ONE,
318: + A( N1*N2 ), N2 )
319: CALL ZPOTRF( 'L', N2, A( N1*N2 ), N2, INFO )
320: IF( INFO.GT.0 )
321: + INFO = INFO + N1
322: *
323: END IF
324: *
325: END IF
326: *
327: ELSE
328: *
329: * N is even
330: *
331: IF( NORMALTRANSR ) THEN
332: *
333: * N is even and TRANSR = 'N'
334: *
335: IF( LOWER ) THEN
336: *
337: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
338: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
339: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
340: *
341: CALL ZPOTRF( 'L', K, A( 1 ), N+1, INFO )
342: IF( INFO.GT.0 )
343: + RETURN
344: CALL ZTRSM( 'R', 'L', 'C', 'N', K, K, CONE, A( 1 ), N+1,
345: + A( K+1 ), N+1 )
346: CALL ZHERK( 'U', 'N', K, K, -ONE, A( K+1 ), N+1, ONE,
347: + A( 0 ), N+1 )
348: CALL ZPOTRF( 'U', K, A( 0 ), N+1, INFO )
349: IF( INFO.GT.0 )
350: + INFO = INFO + K
351: *
352: ELSE
353: *
354: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
355: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
356: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
357: *
358: CALL ZPOTRF( 'L', K, A( K+1 ), N+1, INFO )
359: IF( INFO.GT.0 )
360: + RETURN
361: CALL ZTRSM( 'L', 'L', 'N', 'N', K, K, CONE, A( K+1 ),
362: + N+1, A( 0 ), N+1 )
363: CALL ZHERK( 'U', 'C', K, K, -ONE, A( 0 ), N+1, ONE,
364: + A( K ), N+1 )
365: CALL ZPOTRF( 'U', K, A( K ), N+1, INFO )
366: IF( INFO.GT.0 )
367: + INFO = INFO + K
368: *
369: END IF
370: *
371: ELSE
372: *
373: * N is even and TRANSR = 'C'
374: *
375: IF( LOWER ) THEN
376: *
377: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
378: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
379: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
380: *
381: CALL ZPOTRF( 'U', K, A( 0+K ), K, INFO )
382: IF( INFO.GT.0 )
383: + RETURN
384: CALL ZTRSM( 'L', 'U', 'C', 'N', K, K, CONE, A( K ), N1,
385: + A( K*( K+1 ) ), K )
386: CALL ZHERK( 'L', 'C', K, K, -ONE, A( K*( K+1 ) ), K, ONE,
387: + A( 0 ), K )
388: CALL ZPOTRF( 'L', K, A( 0 ), K, INFO )
389: IF( INFO.GT.0 )
390: + INFO = INFO + K
391: *
392: ELSE
393: *
394: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
395: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
396: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
397: *
398: CALL ZPOTRF( 'U', K, A( K*( K+1 ) ), K, INFO )
399: IF( INFO.GT.0 )
400: + RETURN
401: CALL ZTRSM( 'R', 'U', 'N', 'N', K, K, CONE,
402: + A( K*( K+1 ) ), K, A( 0 ), K )
403: CALL ZHERK( 'L', 'N', K, K, -ONE, A( 0 ), K, ONE,
404: + A( K*K ), K )
405: CALL ZPOTRF( 'L', K, A( K*K ), K, INFO )
406: IF( INFO.GT.0 )
407: + INFO = INFO + K
408: *
409: END IF
410: *
411: END IF
412: *
413: END IF
414: *
415: RETURN
416: *
417: * End of ZPFTRF
418: *
419: END
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