1: *> \brief \b ZTFTRI
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZTFTRI + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztftri.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztftri.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER TRANSR, UPLO, DIAG
25: * INTEGER INFO, N
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 A( 0: * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZTFTRI computes the inverse of a triangular matrix A stored in RFP
38: *> format.
39: *>
40: *> This is a Level 3 BLAS version of the algorithm.
41: *> \endverbatim
42: *
43: * Arguments:
44: * ==========
45: *
46: *> \param[in] TRANSR
47: *> \verbatim
48: *> TRANSR is CHARACTER*1
49: *> = 'N': The Normal TRANSR of RFP A is stored;
50: *> = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
51: *> \endverbatim
52: *>
53: *> \param[in] UPLO
54: *> \verbatim
55: *> UPLO is CHARACTER*1
56: *> = 'U': A is upper triangular;
57: *> = 'L': A is lower triangular.
58: *> \endverbatim
59: *>
60: *> \param[in] DIAG
61: *> \verbatim
62: *> DIAG is CHARACTER*1
63: *> = 'N': A is non-unit triangular;
64: *> = 'U': A is unit triangular.
65: *> \endverbatim
66: *>
67: *> \param[in] N
68: *> \verbatim
69: *> N is INTEGER
70: *> The order of the matrix A. N >= 0.
71: *> \endverbatim
72: *>
73: *> \param[in,out] A
74: *> \verbatim
75: *> A is COMPLEX*16 array, dimension ( N*(N+1)/2 );
76: *> On entry, the triangular matrix A in RFP format. RFP format
77: *> is described by TRANSR, UPLO, and N as follows: If TRANSR =
78: *> 'N' then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
79: *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
80: *> the Conjugate-transpose of RFP A as defined when
81: *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
82: *> follows: If UPLO = 'U' the RFP A contains the nt elements of
83: *> upper packed A; If UPLO = 'L' the RFP A contains the nt
84: *> elements of lower packed A. The LDA of RFP A is (N+1)/2 when
85: *> TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
86: *> even and N is odd. See the Note below for more details.
87: *>
88: *> On exit, the (triangular) inverse of the original matrix, in
89: *> the same storage format.
90: *> \endverbatim
91: *>
92: *> \param[out] INFO
93: *> \verbatim
94: *> INFO is INTEGER
95: *> = 0: successful exit
96: *> < 0: if INFO = -i, the i-th argument had an illegal value
97: *> > 0: if INFO = i, A(i,i) is exactly zero. The triangular
98: *> matrix is singular and its inverse can not be computed.
99: *> \endverbatim
100: *
101: * Authors:
102: * ========
103: *
104: *> \author Univ. of Tennessee
105: *> \author Univ. of California Berkeley
106: *> \author Univ. of Colorado Denver
107: *> \author NAG Ltd.
108: *
109: *> \date December 2016
110: *
111: *> \ingroup complex16OTHERcomputational
112: *
113: *> \par Further Details:
114: * =====================
115: *>
116: *> \verbatim
117: *>
118: *> We first consider Standard Packed Format when N is even.
119: *> We give an example where N = 6.
120: *>
121: *> AP is Upper AP is Lower
122: *>
123: *> 00 01 02 03 04 05 00
124: *> 11 12 13 14 15 10 11
125: *> 22 23 24 25 20 21 22
126: *> 33 34 35 30 31 32 33
127: *> 44 45 40 41 42 43 44
128: *> 55 50 51 52 53 54 55
129: *>
130: *>
131: *> Let TRANSR = 'N'. RFP holds AP as follows:
132: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
133: *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
134: *> conjugate-transpose of the first three columns of AP upper.
135: *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
136: *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
137: *> conjugate-transpose of the last three columns of AP lower.
138: *> To denote conjugate we place -- above the element. This covers the
139: *> case N even and TRANSR = 'N'.
140: *>
141: *> RFP A RFP A
142: *>
143: *> -- -- --
144: *> 03 04 05 33 43 53
145: *> -- --
146: *> 13 14 15 00 44 54
147: *> --
148: *> 23 24 25 10 11 55
149: *>
150: *> 33 34 35 20 21 22
151: *> --
152: *> 00 44 45 30 31 32
153: *> -- --
154: *> 01 11 55 40 41 42
155: *> -- -- --
156: *> 02 12 22 50 51 52
157: *>
158: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
159: *> transpose of RFP A above. One therefore gets:
160: *>
161: *>
162: *> RFP A RFP A
163: *>
164: *> -- -- -- -- -- -- -- -- -- --
165: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
166: *> -- -- -- -- -- -- -- -- -- --
167: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
168: *> -- -- -- -- -- -- -- -- -- --
169: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
170: *>
171: *>
172: *> We next consider Standard Packed Format when N is odd.
173: *> We give an example where N = 5.
174: *>
175: *> AP is Upper AP is Lower
176: *>
177: *> 00 01 02 03 04 00
178: *> 11 12 13 14 10 11
179: *> 22 23 24 20 21 22
180: *> 33 34 30 31 32 33
181: *> 44 40 41 42 43 44
182: *>
183: *>
184: *> Let TRANSR = 'N'. RFP holds AP as follows:
185: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
186: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
187: *> conjugate-transpose of the first two columns of AP upper.
188: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
189: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
190: *> conjugate-transpose of the last two columns of AP lower.
191: *> To denote conjugate we place -- above the element. This covers the
192: *> case N odd and TRANSR = 'N'.
193: *>
194: *> RFP A RFP A
195: *>
196: *> -- --
197: *> 02 03 04 00 33 43
198: *> --
199: *> 12 13 14 10 11 44
200: *>
201: *> 22 23 24 20 21 22
202: *> --
203: *> 00 33 34 30 31 32
204: *> -- --
205: *> 01 11 44 40 41 42
206: *>
207: *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
208: *> transpose of RFP A above. One therefore gets:
209: *>
210: *>
211: *> RFP A RFP A
212: *>
213: *> -- -- -- -- -- -- -- -- --
214: *> 02 12 22 00 01 00 10 20 30 40 50
215: *> -- -- -- -- -- -- -- -- --
216: *> 03 13 23 33 11 33 11 21 31 41 51
217: *> -- -- -- -- -- -- -- -- --
218: *> 04 14 24 34 44 43 44 22 32 42 52
219: *> \endverbatim
220: *>
221: * =====================================================================
222: SUBROUTINE ZTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
223: *
224: * -- LAPACK computational routine (version 3.7.0) --
225: * -- LAPACK is a software package provided by Univ. of Tennessee, --
226: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
227: * December 2016
228: *
229: * .. Scalar Arguments ..
230: CHARACTER TRANSR, UPLO, DIAG
231: INTEGER INFO, N
232: * ..
233: * .. Array Arguments ..
234: COMPLEX*16 A( 0: * )
235: * ..
236: *
237: * =====================================================================
238: *
239: * .. Parameters ..
240: COMPLEX*16 CONE
241: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
242: * ..
243: * .. Local Scalars ..
244: LOGICAL LOWER, NISODD, NORMALTRANSR
245: INTEGER N1, N2, K
246: * ..
247: * .. External Functions ..
248: LOGICAL LSAME
249: EXTERNAL LSAME
250: * ..
251: * .. External Subroutines ..
252: EXTERNAL XERBLA, ZTRMM, ZTRTRI
253: * ..
254: * .. Intrinsic Functions ..
255: INTRINSIC MOD
256: * ..
257: * .. Executable Statements ..
258: *
259: * Test the input parameters.
260: *
261: INFO = 0
262: NORMALTRANSR = LSAME( TRANSR, 'N' )
263: LOWER = LSAME( UPLO, 'L' )
264: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
265: INFO = -1
266: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
267: INFO = -2
268: ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND. .NOT.LSAME( DIAG, 'U' ) )
269: $ THEN
270: INFO = -3
271: ELSE IF( N.LT.0 ) THEN
272: INFO = -4
273: END IF
274: IF( INFO.NE.0 ) THEN
275: CALL XERBLA( 'ZTFTRI', -INFO )
276: RETURN
277: END IF
278: *
279: * Quick return if possible
280: *
281: IF( N.EQ.0 )
282: $ RETURN
283: *
284: * If N is odd, set NISODD = .TRUE.
285: * If N is even, set K = N/2 and NISODD = .FALSE.
286: *
287: IF( MOD( N, 2 ).EQ.0 ) THEN
288: K = N / 2
289: NISODD = .FALSE.
290: ELSE
291: NISODD = .TRUE.
292: END IF
293: *
294: * Set N1 and N2 depending on LOWER
295: *
296: IF( LOWER ) THEN
297: N2 = N / 2
298: N1 = N - N2
299: ELSE
300: N1 = N / 2
301: N2 = N - N1
302: END IF
303: *
304: *
305: * start execution: there are eight cases
306: *
307: IF( NISODD ) THEN
308: *
309: * N is odd
310: *
311: IF( NORMALTRANSR ) THEN
312: *
313: * N is odd and TRANSR = 'N'
314: *
315: IF( LOWER ) THEN
316: *
317: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
318: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
319: * T1 -> a(0), T2 -> a(n), S -> a(n1)
320: *
321: CALL ZTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
322: IF( INFO.GT.0 )
323: $ RETURN
324: CALL ZTRMM( 'R', 'L', 'N', DIAG, N2, N1, -CONE, A( 0 ),
325: $ N, A( N1 ), N )
326: CALL ZTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
327: IF( INFO.GT.0 )
328: $ INFO = INFO + N1
329: IF( INFO.GT.0 )
330: $ RETURN
331: CALL ZTRMM( 'L', 'U', 'C', DIAG, N2, N1, CONE, A( N ), N,
332: $ A( N1 ), N )
333: *
334: ELSE
335: *
336: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
337: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
338: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
339: *
340: CALL ZTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
341: IF( INFO.GT.0 )
342: $ RETURN
343: CALL ZTRMM( 'L', 'L', 'C', DIAG, N1, N2, -CONE, A( N2 ),
344: $ N, A( 0 ), N )
345: CALL ZTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
346: IF( INFO.GT.0 )
347: $ INFO = INFO + N1
348: IF( INFO.GT.0 )
349: $ RETURN
350: CALL ZTRMM( 'R', 'U', 'N', DIAG, N1, N2, CONE, A( N1 ),
351: $ N, A( 0 ), N )
352: *
353: END IF
354: *
355: ELSE
356: *
357: * N is odd and TRANSR = 'C'
358: *
359: IF( LOWER ) THEN
360: *
361: * SRPA for LOWER, TRANSPOSE and N is odd
362: * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
363: *
364: CALL ZTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
365: IF( INFO.GT.0 )
366: $ RETURN
367: CALL ZTRMM( 'L', 'U', 'N', DIAG, N1, N2, -CONE, A( 0 ),
368: $ N1, A( N1*N1 ), N1 )
369: CALL ZTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
370: IF( INFO.GT.0 )
371: $ INFO = INFO + N1
372: IF( INFO.GT.0 )
373: $ RETURN
374: CALL ZTRMM( 'R', 'L', 'C', DIAG, N1, N2, CONE, A( 1 ),
375: $ N1, A( N1*N1 ), N1 )
376: *
377: ELSE
378: *
379: * SRPA for UPPER, TRANSPOSE and N is odd
380: * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
381: *
382: CALL ZTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
383: IF( INFO.GT.0 )
384: $ RETURN
385: CALL ZTRMM( 'R', 'U', 'C', DIAG, N2, N1, -CONE,
386: $ A( N2*N2 ), N2, A( 0 ), N2 )
387: CALL ZTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
388: IF( INFO.GT.0 )
389: $ INFO = INFO + N1
390: IF( INFO.GT.0 )
391: $ RETURN
392: CALL ZTRMM( 'L', 'L', 'N', DIAG, N2, N1, CONE,
393: $ A( N1*N2 ), N2, A( 0 ), N2 )
394: END IF
395: *
396: END IF
397: *
398: ELSE
399: *
400: * N is even
401: *
402: IF( NORMALTRANSR ) THEN
403: *
404: * N is even and TRANSR = 'N'
405: *
406: IF( LOWER ) THEN
407: *
408: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
409: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
410: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
411: *
412: CALL ZTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
413: IF( INFO.GT.0 )
414: $ RETURN
415: CALL ZTRMM( 'R', 'L', 'N', DIAG, K, K, -CONE, A( 1 ),
416: $ N+1, A( K+1 ), N+1 )
417: CALL ZTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
418: IF( INFO.GT.0 )
419: $ INFO = INFO + K
420: IF( INFO.GT.0 )
421: $ RETURN
422: CALL ZTRMM( 'L', 'U', 'C', DIAG, K, K, CONE, A( 0 ), N+1,
423: $ A( K+1 ), N+1 )
424: *
425: ELSE
426: *
427: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
428: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
429: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
430: *
431: CALL ZTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
432: IF( INFO.GT.0 )
433: $ RETURN
434: CALL ZTRMM( 'L', 'L', 'C', DIAG, K, K, -CONE, A( K+1 ),
435: $ N+1, A( 0 ), N+1 )
436: CALL ZTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
437: IF( INFO.GT.0 )
438: $ INFO = INFO + K
439: IF( INFO.GT.0 )
440: $ RETURN
441: CALL ZTRMM( 'R', 'U', 'N', DIAG, K, K, CONE, A( K ), N+1,
442: $ A( 0 ), N+1 )
443: END IF
444: ELSE
445: *
446: * N is even and TRANSR = 'C'
447: *
448: IF( LOWER ) THEN
449: *
450: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
451: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
452: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
453: *
454: CALL ZTRTRI( 'U', DIAG, K, A( K ), K, INFO )
455: IF( INFO.GT.0 )
456: $ RETURN
457: CALL ZTRMM( 'L', 'U', 'N', DIAG, K, K, -CONE, A( K ), K,
458: $ A( K*( K+1 ) ), K )
459: CALL ZTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
460: IF( INFO.GT.0 )
461: $ INFO = INFO + K
462: IF( INFO.GT.0 )
463: $ RETURN
464: CALL ZTRMM( 'R', 'L', 'C', DIAG, K, K, CONE, A( 0 ), K,
465: $ A( K*( K+1 ) ), K )
466: ELSE
467: *
468: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
469: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
470: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
471: *
472: CALL ZTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
473: IF( INFO.GT.0 )
474: $ RETURN
475: CALL ZTRMM( 'R', 'U', 'C', DIAG, K, K, -CONE,
476: $ A( K*( K+1 ) ), K, A( 0 ), K )
477: CALL ZTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
478: IF( INFO.GT.0 )
479: $ INFO = INFO + K
480: IF( INFO.GT.0 )
481: $ RETURN
482: CALL ZTRMM( 'L', 'L', 'N', DIAG, K, K, CONE, A( K*K ), K,
483: $ A( 0 ), K )
484: END IF
485: END IF
486: END IF
487: *
488: RETURN
489: *
490: * End of ZTFTRI
491: *
492: END
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