1: *> \brief \b ZPFTRI
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER TRANSR, UPLO
25: * INTEGER INFO, N
26: * .. Array Arguments ..
27: * COMPLEX*16 A( 0: * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZPFTRI computes the inverse of a complex Hermitian positive definite
37: *> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
38: *> computed by ZPFTRF.
39: *> \endverbatim
40: *
41: * Arguments:
42: * ==========
43: *
44: *> \param[in] TRANSR
45: *> \verbatim
46: *> TRANSR is CHARACTER*1
47: *> = 'N': The Normal TRANSR of RFP A is stored;
48: *> = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
49: *> \endverbatim
50: *>
51: *> \param[in] UPLO
52: *> \verbatim
53: *> UPLO is CHARACTER*1
54: *> = 'U': Upper triangle of A is stored;
55: *> = 'L': Lower triangle of A is stored.
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The order of the matrix A. N >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in,out] A
65: *> \verbatim
66: *> A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
67: *> On entry, the Hermitian matrix A in RFP format. RFP format is
68: *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
69: *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
70: *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
71: *> the Conjugate-transpose of RFP A as defined when
72: *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
73: *> follows: If UPLO = 'U' the RFP A contains the nt elements of
74: *> upper packed A. If UPLO = 'L' the RFP A contains the elements
75: *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
76: *> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
77: *> is odd. See the Note below for more details.
78: *>
79: *> On exit, the Hermitian inverse of the original matrix, in the
80: *> same storage format.
81: *> \endverbatim
82: *>
83: *> \param[out] INFO
84: *> \verbatim
85: *> INFO is INTEGER
86: *> = 0: successful exit
87: *> < 0: if INFO = -i, the i-th argument had an illegal value
88: *> > 0: if INFO = i, the (i,i) element of the factor U or L is
89: *> zero, and the inverse could not be computed.
90: *> \endverbatim
91: *
92: * Authors:
93: * ========
94: *
95: *> \author Univ. of Tennessee
96: *> \author Univ. of California Berkeley
97: *> \author Univ. of Colorado Denver
98: *> \author NAG Ltd.
99: *
100: *> \ingroup complex16OTHERcomputational
101: *
102: *> \par Further Details:
103: * =====================
104: *>
105: *> \verbatim
106: *>
107: *> We first consider Standard Packed Format when N is even.
108: *> We give an example where N = 6.
109: *>
110: *> AP is Upper AP is Lower
111: *>
112: *> 00 01 02 03 04 05 00
113: *> 11 12 13 14 15 10 11
114: *> 22 23 24 25 20 21 22
115: *> 33 34 35 30 31 32 33
116: *> 44 45 40 41 42 43 44
117: *> 55 50 51 52 53 54 55
118: *>
119: *>
120: *> Let TRANSR = 'N'. RFP holds AP as follows:
121: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
122: *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
123: *> conjugate-transpose of the first three columns of AP upper.
124: *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
125: *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
126: *> conjugate-transpose of the last three columns of AP lower.
127: *> To denote conjugate we place -- above the element. This covers the
128: *> case N even and TRANSR = 'N'.
129: *>
130: *> RFP A RFP A
131: *>
132: *> -- -- --
133: *> 03 04 05 33 43 53
134: *> -- --
135: *> 13 14 15 00 44 54
136: *> --
137: *> 23 24 25 10 11 55
138: *>
139: *> 33 34 35 20 21 22
140: *> --
141: *> 00 44 45 30 31 32
142: *> -- --
143: *> 01 11 55 40 41 42
144: *> -- -- --
145: *> 02 12 22 50 51 52
146: *>
147: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
148: *> transpose of RFP A above. One therefore gets:
149: *>
150: *>
151: *> RFP A RFP A
152: *>
153: *> -- -- -- -- -- -- -- -- -- --
154: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
155: *> -- -- -- -- -- -- -- -- -- --
156: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
157: *> -- -- -- -- -- -- -- -- -- --
158: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
159: *>
160: *>
161: *> We next consider Standard Packed Format when N is odd.
162: *> We give an example where N = 5.
163: *>
164: *> AP is Upper AP is Lower
165: *>
166: *> 00 01 02 03 04 00
167: *> 11 12 13 14 10 11
168: *> 22 23 24 20 21 22
169: *> 33 34 30 31 32 33
170: *> 44 40 41 42 43 44
171: *>
172: *>
173: *> Let TRANSR = 'N'. RFP holds AP as follows:
174: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
175: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
176: *> conjugate-transpose of the first two columns of AP upper.
177: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
178: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
179: *> conjugate-transpose of the last two columns of AP lower.
180: *> To denote conjugate we place -- above the element. This covers the
181: *> case N odd and TRANSR = 'N'.
182: *>
183: *> RFP A RFP A
184: *>
185: *> -- --
186: *> 02 03 04 00 33 43
187: *> --
188: *> 12 13 14 10 11 44
189: *>
190: *> 22 23 24 20 21 22
191: *> --
192: *> 00 33 34 30 31 32
193: *> -- --
194: *> 01 11 44 40 41 42
195: *>
196: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
197: *> transpose of RFP A above. One therefore gets:
198: *>
199: *>
200: *> RFP A RFP A
201: *>
202: *> -- -- -- -- -- -- -- -- --
203: *> 02 12 22 00 01 00 10 20 30 40 50
204: *> -- -- -- -- -- -- -- -- --
205: *> 03 13 23 33 11 33 11 21 31 41 51
206: *> -- -- -- -- -- -- -- -- --
207: *> 04 14 24 34 44 43 44 22 32 42 52
208: *> \endverbatim
209: *>
210: * =====================================================================
211: SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
212: *
213: * -- LAPACK computational routine --
214: * -- LAPACK is a software package provided by Univ. of Tennessee, --
215: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216: *
217: * .. Scalar Arguments ..
218: CHARACTER TRANSR, UPLO
219: INTEGER INFO, N
220: * .. Array Arguments ..
221: COMPLEX*16 A( 0: * )
222: * ..
223: *
224: * =====================================================================
225: *
226: * .. Parameters ..
227: DOUBLE PRECISION ONE
228: COMPLEX*16 CONE
229: PARAMETER ( ONE = 1.D0, CONE = ( 1.D0, 0.D0 ) )
230: * ..
231: * .. Local Scalars ..
232: LOGICAL LOWER, NISODD, NORMALTRANSR
233: INTEGER N1, N2, K
234: * ..
235: * .. External Functions ..
236: LOGICAL LSAME
237: EXTERNAL LSAME
238: * ..
239: * .. External Subroutines ..
240: EXTERNAL XERBLA, ZTFTRI, ZLAUUM, ZTRMM, ZHERK
241: * ..
242: * .. Intrinsic Functions ..
243: INTRINSIC MOD
244: * ..
245: * .. Executable Statements ..
246: *
247: * Test the input parameters.
248: *
249: INFO = 0
250: NORMALTRANSR = LSAME( TRANSR, 'N' )
251: LOWER = LSAME( UPLO, 'L' )
252: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
253: INFO = -1
254: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
255: INFO = -2
256: ELSE IF( N.LT.0 ) THEN
257: INFO = -3
258: END IF
259: IF( INFO.NE.0 ) THEN
260: CALL XERBLA( 'ZPFTRI', -INFO )
261: RETURN
262: END IF
263: *
264: * Quick return if possible
265: *
266: IF( N.EQ.0 )
267: $ RETURN
268: *
269: * Invert the triangular Cholesky factor U or L.
270: *
271: CALL ZTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
272: IF( INFO.GT.0 )
273: $ RETURN
274: *
275: * If N is odd, set NISODD = .TRUE.
276: * If N is even, set K = N/2 and NISODD = .FALSE.
277: *
278: IF( MOD( N, 2 ).EQ.0 ) THEN
279: K = N / 2
280: NISODD = .FALSE.
281: ELSE
282: NISODD = .TRUE.
283: END IF
284: *
285: * Set N1 and N2 depending on LOWER
286: *
287: IF( LOWER ) THEN
288: N2 = N / 2
289: N1 = N - N2
290: ELSE
291: N1 = N / 2
292: N2 = N - N1
293: END IF
294: *
295: * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
296: * inv(L)^C*inv(L). There are eight cases.
297: *
298: IF( NISODD ) THEN
299: *
300: * N is odd
301: *
302: IF( NORMALTRANSR ) THEN
303: *
304: * N is odd and TRANSR = 'N'
305: *
306: IF( LOWER ) THEN
307: *
308: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
309: * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
310: * T1 -> a(0), T2 -> a(n), S -> a(N1)
311: *
312: CALL ZLAUUM( 'L', N1, A( 0 ), N, INFO )
313: CALL ZHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE,
314: $ A( 0 ), N )
315: CALL ZTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N,
316: $ A( N1 ), N )
317: CALL ZLAUUM( 'U', N2, A( N ), N, INFO )
318: *
319: ELSE
320: *
321: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
322: * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
323: * T1 -> a(N2), T2 -> a(N1), S -> a(0)
324: *
325: CALL ZLAUUM( 'L', N1, A( N2 ), N, INFO )
326: CALL ZHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
327: $ A( N2 ), N )
328: CALL ZTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N,
329: $ A( 0 ), N )
330: CALL ZLAUUM( 'U', N2, A( N1 ), N, INFO )
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), T2 -> a(1), S -> a(0+N1*N1)
342: *
343: CALL ZLAUUM( 'U', N1, A( 0 ), N1, INFO )
344: CALL ZHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
345: $ A( 0 ), N1 )
346: CALL ZTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1,
347: $ A( N1*N1 ), N1 )
348: CALL ZLAUUM( 'L', N2, A( 1 ), N1, INFO )
349: *
350: ELSE
351: *
352: * SRPA for UPPER, TRANSPOSE, and N is odd
353: * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
354: *
355: CALL ZLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
356: CALL ZHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE,
357: $ A( N2*N2 ), N2 )
358: CALL ZTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ),
359: $ N2, A( 0 ), N2 )
360: CALL ZLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
361: *
362: END IF
363: *
364: END IF
365: *
366: ELSE
367: *
368: * N is even
369: *
370: IF( NORMALTRANSR ) THEN
371: *
372: * N is even and TRANSR = 'N'
373: *
374: IF( LOWER ) THEN
375: *
376: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
377: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
378: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
379: *
380: CALL ZLAUUM( 'L', K, A( 1 ), N+1, INFO )
381: CALL ZHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE,
382: $ A( 1 ), N+1 )
383: CALL ZTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1,
384: $ A( K+1 ), N+1 )
385: CALL ZLAUUM( 'U', K, A( 0 ), N+1, INFO )
386: *
387: ELSE
388: *
389: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
390: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
391: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
392: *
393: CALL ZLAUUM( 'L', K, A( K+1 ), N+1, INFO )
394: CALL ZHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
395: $ A( K+1 ), N+1 )
396: CALL ZTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1,
397: $ A( 0 ), N+1 )
398: CALL ZLAUUM( 'U', K, A( K ), N+1, INFO )
399: *
400: END IF
401: *
402: ELSE
403: *
404: * N is even and TRANSR = 'C'
405: *
406: IF( LOWER ) THEN
407: *
408: * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
409: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
410: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
411: *
412: CALL ZLAUUM( 'U', K, A( K ), K, INFO )
413: CALL ZHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
414: $ A( K ), K )
415: CALL ZTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K,
416: $ A( K*( K+1 ) ), K )
417: CALL ZLAUUM( 'L', K, A( 0 ), K, INFO )
418: *
419: ELSE
420: *
421: * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
422: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
423: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
424: *
425: CALL ZLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
426: CALL ZHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE,
427: $ A( K*( K+1 ) ), K )
428: CALL ZTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K,
429: $ A( 0 ), K )
430: CALL ZLAUUM( 'L', K, A( K*K ), K, INFO )
431: *
432: END IF
433: *
434: END IF
435: *
436: END IF
437: *
438: RETURN
439: *
440: * End of ZPFTRI
441: *
442: END
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