1: *> \brief \b DTFTRI
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER TRANSR, UPLO, DIAG
25: * INTEGER INFO, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( 0: * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DTFTRI 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: *> = 'T': The 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 DOUBLE PRECISION array, dimension (0:nt-1);
76: *> nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
77: *> Positive Definite matrix A in RFP format. RFP format is
78: *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
79: *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
80: *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
81: *> the transpose of RFP A as defined when
82: *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
83: *> follows: If UPLO = 'U' the RFP A contains the nt elements of
84: *> upper packed A; If UPLO = 'L' the RFP A contains the nt
85: *> elements of lower packed A. The LDA of RFP A is (N+1)/2 when
86: *> TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
87: *> even and N is odd. See the Note below for more details.
88: *>
89: *> On exit, the (triangular) inverse of the original matrix, in
90: *> the same storage format.
91: *> \endverbatim
92: *>
93: *> \param[out] INFO
94: *> \verbatim
95: *> INFO is INTEGER
96: *> = 0: successful exit
97: *> < 0: if INFO = -i, the i-th argument had an illegal value
98: *> > 0: if INFO = i, A(i,i) is exactly zero. The triangular
99: *> matrix is singular and its inverse can not be computed.
100: *> \endverbatim
101: *
102: * Authors:
103: * ========
104: *
105: *> \author Univ. of Tennessee
106: *> \author Univ. of California Berkeley
107: *> \author Univ. of Colorado Denver
108: *> \author NAG Ltd.
109: *
110: *> \date November 2011
111: *
112: *> \ingroup doubleOTHERcomputational
113: *
114: *> \par Further Details:
115: * =====================
116: *>
117: *> \verbatim
118: *>
119: *> We first consider Rectangular Full Packed (RFP) Format when N is
120: *> even. We give an example where N = 6.
121: *>
122: *> AP is Upper AP is Lower
123: *>
124: *> 00 01 02 03 04 05 00
125: *> 11 12 13 14 15 10 11
126: *> 22 23 24 25 20 21 22
127: *> 33 34 35 30 31 32 33
128: *> 44 45 40 41 42 43 44
129: *> 55 50 51 52 53 54 55
130: *>
131: *>
132: *> Let TRANSR = 'N'. RFP holds AP as follows:
133: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
134: *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
135: *> the transpose of the first three columns of AP upper.
136: *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
137: *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
138: *> the transpose of the last three columns of AP lower.
139: *> This covers the case N even and TRANSR = 'N'.
140: *>
141: *> RFP A RFP A
142: *>
143: *> 03 04 05 33 43 53
144: *> 13 14 15 00 44 54
145: *> 23 24 25 10 11 55
146: *> 33 34 35 20 21 22
147: *> 00 44 45 30 31 32
148: *> 01 11 55 40 41 42
149: *> 02 12 22 50 51 52
150: *>
151: *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
152: *> transpose of RFP A above. One therefore gets:
153: *>
154: *>
155: *> RFP A RFP A
156: *>
157: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
158: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
159: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
160: *>
161: *>
162: *> We then consider Rectangular Full Packed (RFP) Format when N is
163: *> odd. We give an example where N = 5.
164: *>
165: *> AP is Upper AP is Lower
166: *>
167: *> 00 01 02 03 04 00
168: *> 11 12 13 14 10 11
169: *> 22 23 24 20 21 22
170: *> 33 34 30 31 32 33
171: *> 44 40 41 42 43 44
172: *>
173: *>
174: *> Let TRANSR = 'N'. RFP holds AP as follows:
175: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
176: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
177: *> the transpose of the first two columns of AP upper.
178: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
179: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
180: *> the transpose of the last two columns of AP lower.
181: *> This covers the case N odd and TRANSR = 'N'.
182: *>
183: *> RFP A RFP A
184: *>
185: *> 02 03 04 00 33 43
186: *> 12 13 14 10 11 44
187: *> 22 23 24 20 21 22
188: *> 00 33 34 30 31 32
189: *> 01 11 44 40 41 42
190: *>
191: *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
192: *> transpose of RFP A above. One therefore gets:
193: *>
194: *> RFP A RFP A
195: *>
196: *> 02 12 22 00 01 00 10 20 30 40 50
197: *> 03 13 23 33 11 33 11 21 31 41 51
198: *> 04 14 24 34 44 43 44 22 32 42 52
199: *> \endverbatim
200: *>
201: * =====================================================================
202: SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
203: *
204: * -- LAPACK computational routine (version 3.4.0) --
205: * -- LAPACK is a software package provided by Univ. of Tennessee, --
206: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207: * November 2011
208: *
209: * .. Scalar Arguments ..
210: CHARACTER TRANSR, UPLO, DIAG
211: INTEGER INFO, N
212: * ..
213: * .. Array Arguments ..
214: DOUBLE PRECISION A( 0: * )
215: * ..
216: *
217: * =====================================================================
218: *
219: * .. Parameters ..
220: DOUBLE PRECISION ONE
221: PARAMETER ( ONE = 1.0D+0 )
222: * ..
223: * .. Local Scalars ..
224: LOGICAL LOWER, NISODD, NORMALTRANSR
225: INTEGER N1, N2, K
226: * ..
227: * .. External Functions ..
228: LOGICAL LSAME
229: EXTERNAL LSAME
230: * ..
231: * .. External Subroutines ..
232: EXTERNAL XERBLA, DTRMM, DTRTRI
233: * ..
234: * .. Intrinsic Functions ..
235: INTRINSIC MOD
236: * ..
237: * .. Executable Statements ..
238: *
239: * Test the input parameters.
240: *
241: INFO = 0
242: NORMALTRANSR = LSAME( TRANSR, 'N' )
243: LOWER = LSAME( UPLO, 'L' )
244: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
245: INFO = -1
246: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
247: INFO = -2
248: ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND. .NOT.LSAME( DIAG, 'U' ) )
249: $ THEN
250: INFO = -3
251: ELSE IF( N.LT.0 ) THEN
252: INFO = -4
253: END IF
254: IF( INFO.NE.0 ) THEN
255: CALL XERBLA( 'DTFTRI', -INFO )
256: RETURN
257: END IF
258: *
259: * Quick return if possible
260: *
261: IF( N.EQ.0 )
262: $ RETURN
263: *
264: * If N is odd, set NISODD = .TRUE.
265: * If N is even, set K = N/2 and NISODD = .FALSE.
266: *
267: IF( MOD( N, 2 ).EQ.0 ) THEN
268: K = N / 2
269: NISODD = .FALSE.
270: ELSE
271: NISODD = .TRUE.
272: END IF
273: *
274: * Set N1 and N2 depending on LOWER
275: *
276: IF( LOWER ) THEN
277: N2 = N / 2
278: N1 = N - N2
279: ELSE
280: N1 = N / 2
281: N2 = N - N1
282: END IF
283: *
284: *
285: * start execution: there are eight cases
286: *
287: IF( NISODD ) THEN
288: *
289: * N is odd
290: *
291: IF( NORMALTRANSR ) THEN
292: *
293: * N is odd and TRANSR = 'N'
294: *
295: IF( LOWER ) THEN
296: *
297: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
298: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
299: * T1 -> a(0), T2 -> a(n), S -> a(n1)
300: *
301: CALL DTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
302: IF( INFO.GT.0 )
303: $ RETURN
304: CALL DTRMM( 'R', 'L', 'N', DIAG, N2, N1, -ONE, A( 0 ),
305: $ N, A( N1 ), N )
306: CALL DTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
307: IF( INFO.GT.0 )
308: $ INFO = INFO + N1
309: IF( INFO.GT.0 )
310: $ RETURN
311: CALL DTRMM( 'L', 'U', 'T', DIAG, N2, N1, ONE, A( N ), N,
312: $ A( N1 ), N )
313: *
314: ELSE
315: *
316: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
317: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
318: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
319: *
320: CALL DTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
321: IF( INFO.GT.0 )
322: $ RETURN
323: CALL DTRMM( 'L', 'L', 'T', DIAG, N1, N2, -ONE, A( N2 ),
324: $ N, A( 0 ), N )
325: CALL DTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
326: IF( INFO.GT.0 )
327: $ INFO = INFO + N1
328: IF( INFO.GT.0 )
329: $ RETURN
330: CALL DTRMM( 'R', 'U', 'N', DIAG, N1, N2, ONE, A( N1 ),
331: $ N, A( 0 ), N )
332: *
333: END IF
334: *
335: ELSE
336: *
337: * N is odd and TRANSR = 'T'
338: *
339: IF( LOWER ) THEN
340: *
341: * SRPA for LOWER, TRANSPOSE and N is odd
342: * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
343: *
344: CALL DTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
345: IF( INFO.GT.0 )
346: $ RETURN
347: CALL DTRMM( 'L', 'U', 'N', DIAG, N1, N2, -ONE, A( 0 ),
348: $ N1, A( N1*N1 ), N1 )
349: CALL DTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
350: IF( INFO.GT.0 )
351: $ INFO = INFO + N1
352: IF( INFO.GT.0 )
353: $ RETURN
354: CALL DTRMM( 'R', 'L', 'T', DIAG, N1, N2, ONE, A( 1 ),
355: $ N1, A( N1*N1 ), N1 )
356: *
357: ELSE
358: *
359: * SRPA for UPPER, TRANSPOSE and N is odd
360: * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
361: *
362: CALL DTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
363: IF( INFO.GT.0 )
364: $ RETURN
365: CALL DTRMM( 'R', 'U', 'T', DIAG, N2, N1, -ONE,
366: $ A( N2*N2 ), N2, A( 0 ), N2 )
367: CALL DTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
368: IF( INFO.GT.0 )
369: $ INFO = INFO + N1
370: IF( INFO.GT.0 )
371: $ RETURN
372: CALL DTRMM( 'L', 'L', 'N', DIAG, N2, N1, ONE,
373: $ A( N1*N2 ), N2, A( 0 ), N2 )
374: END IF
375: *
376: END IF
377: *
378: ELSE
379: *
380: * N is even
381: *
382: IF( NORMALTRANSR ) THEN
383: *
384: * N is even and TRANSR = 'N'
385: *
386: IF( LOWER ) THEN
387: *
388: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
389: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
390: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
391: *
392: CALL DTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
393: IF( INFO.GT.0 )
394: $ RETURN
395: CALL DTRMM( 'R', 'L', 'N', DIAG, K, K, -ONE, A( 1 ),
396: $ N+1, A( K+1 ), N+1 )
397: CALL DTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
398: IF( INFO.GT.0 )
399: $ INFO = INFO + K
400: IF( INFO.GT.0 )
401: $ RETURN
402: CALL DTRMM( 'L', 'U', 'T', DIAG, K, K, ONE, A( 0 ), N+1,
403: $ A( K+1 ), N+1 )
404: *
405: ELSE
406: *
407: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
408: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
409: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
410: *
411: CALL DTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
412: IF( INFO.GT.0 )
413: $ RETURN
414: CALL DTRMM( 'L', 'L', 'T', DIAG, K, K, -ONE, A( K+1 ),
415: $ N+1, A( 0 ), N+1 )
416: CALL DTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
417: IF( INFO.GT.0 )
418: $ INFO = INFO + K
419: IF( INFO.GT.0 )
420: $ RETURN
421: CALL DTRMM( 'R', 'U', 'N', DIAG, K, K, ONE, A( K ), N+1,
422: $ A( 0 ), N+1 )
423: END IF
424: ELSE
425: *
426: * N is even and TRANSR = 'T'
427: *
428: IF( LOWER ) THEN
429: *
430: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
431: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
432: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
433: *
434: CALL DTRTRI( 'U', DIAG, K, A( K ), K, INFO )
435: IF( INFO.GT.0 )
436: $ RETURN
437: CALL DTRMM( 'L', 'U', 'N', DIAG, K, K, -ONE, A( K ), K,
438: $ A( K*( K+1 ) ), K )
439: CALL DTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
440: IF( INFO.GT.0 )
441: $ INFO = INFO + K
442: IF( INFO.GT.0 )
443: $ RETURN
444: CALL DTRMM( 'R', 'L', 'T', DIAG, K, K, ONE, A( 0 ), K,
445: $ A( K*( K+1 ) ), K )
446: ELSE
447: *
448: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
449: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
450: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
451: *
452: CALL DTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
453: IF( INFO.GT.0 )
454: $ RETURN
455: CALL DTRMM( 'R', 'U', 'T', DIAG, K, K, -ONE,
456: $ A( K*( K+1 ) ), K, A( 0 ), K )
457: CALL DTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
458: IF( INFO.GT.0 )
459: $ INFO = INFO + K
460: IF( INFO.GT.0 )
461: $ RETURN
462: CALL DTRMM( 'L', 'L', 'N', DIAG, K, K, ONE, A( K*K ), K,
463: $ A( 0 ), K )
464: END IF
465: END IF
466: END IF
467: *
468: RETURN
469: *
470: * End of DTFTRI
471: *
472: END
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