1: SUBROUTINE ZTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
2: *
3: * -- LAPACK routine (version 3.3.0) --
4: *
5: * -- Contributed by Fred Gustavson of the IBM Watson Research Center --
6: * November 2010
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: * .. Scalar Arguments ..
12: CHARACTER TRANSR, UPLO, DIAG
13: INTEGER INFO, N
14: * ..
15: * .. Array Arguments ..
16: COMPLEX*16 A( 0: * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * ZTFTRI computes the inverse of a triangular matrix A stored in RFP
23: * format.
24: *
25: * This is a Level 3 BLAS version of the algorithm.
26: *
27: * Arguments
28: * =========
29: *
30: * TRANSR (input) CHARACTER*1
31: * = 'N': The Normal TRANSR of RFP A is stored;
32: * = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
33: *
34: * UPLO (input) CHARACTER*1
35: * = 'U': A is upper triangular;
36: * = 'L': A is lower triangular.
37: *
38: * DIAG (input) CHARACTER*1
39: * = 'N': A is non-unit triangular;
40: * = 'U': A is unit triangular.
41: *
42: * N (input) INTEGER
43: * The order of the matrix A. N >= 0.
44: *
45: * A (input/output) COMPLEX*16 array, dimension ( N*(N+1)/2 );
46: * On entry, the triangular matrix A in RFP format. RFP format
47: * is described by TRANSR, UPLO, and N as follows: If TRANSR =
48: * 'N' then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
49: * (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
50: * the Conjugate-transpose of RFP A as defined when
51: * TRANSR = 'N'. The contents of RFP A are defined by UPLO as
52: * follows: If UPLO = 'U' the RFP A contains the nt elements of
53: * upper packed A; If UPLO = 'L' the RFP A contains the nt
54: * elements of lower packed A. The LDA of RFP A is (N+1)/2 when
55: * TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
56: * even and N is odd. See the Note below for more details.
57: *
58: * On exit, the (triangular) inverse of the original matrix, in
59: * the same storage format.
60: *
61: * INFO (output) INTEGER
62: * = 0: successful exit
63: * < 0: if INFO = -i, the i-th argument had an illegal value
64: * > 0: if INFO = i, A(i,i) is exactly zero. The triangular
65: * matrix is singular and its inverse can not be computed.
66: *
67: * Further Details
68: * ===============
69: *
70: * We first consider Standard Packed Format when N is even.
71: * We give an example where N = 6.
72: *
73: * AP is Upper AP is Lower
74: *
75: * 00 01 02 03 04 05 00
76: * 11 12 13 14 15 10 11
77: * 22 23 24 25 20 21 22
78: * 33 34 35 30 31 32 33
79: * 44 45 40 41 42 43 44
80: * 55 50 51 52 53 54 55
81: *
82: *
83: * Let TRANSR = 'N'. RFP holds AP as follows:
84: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
85: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
86: * conjugate-transpose of the first three columns of AP upper.
87: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
88: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
89: * conjugate-transpose of the last three columns of AP lower.
90: * To denote conjugate we place -- above the element. This covers the
91: * case N even and TRANSR = 'N'.
92: *
93: * RFP A RFP A
94: *
95: * -- -- --
96: * 03 04 05 33 43 53
97: * -- --
98: * 13 14 15 00 44 54
99: * --
100: * 23 24 25 10 11 55
101: *
102: * 33 34 35 20 21 22
103: * --
104: * 00 44 45 30 31 32
105: * -- --
106: * 01 11 55 40 41 42
107: * -- -- --
108: * 02 12 22 50 51 52
109: *
110: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
111: * transpose of RFP A above. One therefore gets:
112: *
113: *
114: * RFP A RFP A
115: *
116: * -- -- -- -- -- -- -- -- -- --
117: * 03 13 23 33 00 01 02 33 00 10 20 30 40 50
118: * -- -- -- -- -- -- -- -- -- --
119: * 04 14 24 34 44 11 12 43 44 11 21 31 41 51
120: * -- -- -- -- -- -- -- -- -- --
121: * 05 15 25 35 45 55 22 53 54 55 22 32 42 52
122: *
123: *
124: * We next consider Standard Packed Format when N is odd.
125: * We give an example where N = 5.
126: *
127: * AP is Upper AP is Lower
128: *
129: * 00 01 02 03 04 00
130: * 11 12 13 14 10 11
131: * 22 23 24 20 21 22
132: * 33 34 30 31 32 33
133: * 44 40 41 42 43 44
134: *
135: *
136: * Let TRANSR = 'N'. RFP holds AP as follows:
137: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
138: * three columns of AP upper. The lower triangle A(3:4,0:1) consists of
139: * conjugate-transpose of the first two columns of AP upper.
140: * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
141: * three columns of AP lower. The upper triangle A(0:1,1:2) consists of
142: * conjugate-transpose of the last two columns of AP lower.
143: * To denote conjugate we place -- above the element. This covers the
144: * case N odd and TRANSR = 'N'.
145: *
146: * RFP A RFP A
147: *
148: * -- --
149: * 02 03 04 00 33 43
150: * --
151: * 12 13 14 10 11 44
152: *
153: * 22 23 24 20 21 22
154: * --
155: * 00 33 34 30 31 32
156: * -- --
157: * 01 11 44 40 41 42
158: *
159: * Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
160: * transpose of RFP A above. One therefore gets:
161: *
162: *
163: * RFP A RFP A
164: *
165: * -- -- -- -- -- -- -- -- --
166: * 02 12 22 00 01 00 10 20 30 40 50
167: * -- -- -- -- -- -- -- -- --
168: * 03 13 23 33 11 33 11 21 31 41 51
169: * -- -- -- -- -- -- -- -- --
170: * 04 14 24 34 44 43 44 22 32 42 52
171: *
172: * =====================================================================
173: *
174: * .. Parameters ..
175: COMPLEX*16 CONE
176: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
177: * ..
178: * .. Local Scalars ..
179: LOGICAL LOWER, NISODD, NORMALTRANSR
180: INTEGER N1, N2, K
181: * ..
182: * .. External Functions ..
183: LOGICAL LSAME
184: EXTERNAL LSAME
185: * ..
186: * .. External Subroutines ..
187: EXTERNAL XERBLA, ZTRMM, ZTRTRI
188: * ..
189: * .. Intrinsic Functions ..
190: INTRINSIC MOD
191: * ..
192: * .. Executable Statements ..
193: *
194: * Test the input parameters.
195: *
196: INFO = 0
197: NORMALTRANSR = LSAME( TRANSR, 'N' )
198: LOWER = LSAME( UPLO, 'L' )
199: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
200: INFO = -1
201: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
202: INFO = -2
203: ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND. .NOT.LSAME( DIAG, 'U' ) )
204: + THEN
205: INFO = -3
206: ELSE IF( N.LT.0 ) THEN
207: INFO = -4
208: END IF
209: IF( INFO.NE.0 ) THEN
210: CALL XERBLA( 'ZTFTRI', -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: *
240: * start execution: there are eight cases
241: *
242: IF( NISODD ) THEN
243: *
244: * N is odd
245: *
246: IF( NORMALTRANSR ) THEN
247: *
248: * N is odd and TRANSR = 'N'
249: *
250: IF( LOWER ) THEN
251: *
252: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
253: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
254: * T1 -> a(0), T2 -> a(n), S -> a(n1)
255: *
256: CALL ZTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
257: IF( INFO.GT.0 )
258: + RETURN
259: CALL ZTRMM( 'R', 'L', 'N', DIAG, N2, N1, -CONE, A( 0 ),
260: + N, A( N1 ), N )
261: CALL ZTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
262: IF( INFO.GT.0 )
263: + INFO = INFO + N1
264: IF( INFO.GT.0 )
265: + RETURN
266: CALL ZTRMM( 'L', 'U', 'C', DIAG, N2, N1, CONE, A( N ), N,
267: + A( N1 ), N )
268: *
269: ELSE
270: *
271: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
272: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
273: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
274: *
275: CALL ZTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
276: IF( INFO.GT.0 )
277: + RETURN
278: CALL ZTRMM( 'L', 'L', 'C', DIAG, N1, N2, -CONE, A( N2 ),
279: + N, A( 0 ), N )
280: CALL ZTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
281: IF( INFO.GT.0 )
282: + INFO = INFO + N1
283: IF( INFO.GT.0 )
284: + RETURN
285: CALL ZTRMM( 'R', 'U', 'N', DIAG, N1, N2, CONE, A( N1 ),
286: + N, A( 0 ), N )
287: *
288: END IF
289: *
290: ELSE
291: *
292: * N is odd and TRANSR = 'C'
293: *
294: IF( LOWER ) THEN
295: *
296: * SRPA for LOWER, TRANSPOSE and N is odd
297: * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
298: *
299: CALL ZTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
300: IF( INFO.GT.0 )
301: + RETURN
302: CALL ZTRMM( 'L', 'U', 'N', DIAG, N1, N2, -CONE, A( 0 ),
303: + N1, A( N1*N1 ), N1 )
304: CALL ZTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
305: IF( INFO.GT.0 )
306: + INFO = INFO + N1
307: IF( INFO.GT.0 )
308: + RETURN
309: CALL ZTRMM( 'R', 'L', 'C', DIAG, N1, N2, CONE, A( 1 ),
310: + N1, A( N1*N1 ), N1 )
311: *
312: ELSE
313: *
314: * SRPA for UPPER, TRANSPOSE and N is odd
315: * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
316: *
317: CALL ZTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
318: IF( INFO.GT.0 )
319: + RETURN
320: CALL ZTRMM( 'R', 'U', 'C', DIAG, N2, N1, -CONE,
321: + A( N2*N2 ), N2, A( 0 ), N2 )
322: CALL ZTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
323: IF( INFO.GT.0 )
324: + INFO = INFO + N1
325: IF( INFO.GT.0 )
326: + RETURN
327: CALL ZTRMM( 'L', 'L', 'N', DIAG, N2, N1, CONE,
328: + A( N1*N2 ), N2, A( 0 ), N2 )
329: END IF
330: *
331: END IF
332: *
333: ELSE
334: *
335: * N is even
336: *
337: IF( NORMALTRANSR ) THEN
338: *
339: * N is even and TRANSR = 'N'
340: *
341: IF( LOWER ) THEN
342: *
343: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
344: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
345: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
346: *
347: CALL ZTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
348: IF( INFO.GT.0 )
349: + RETURN
350: CALL ZTRMM( 'R', 'L', 'N', DIAG, K, K, -CONE, A( 1 ),
351: + N+1, A( K+1 ), N+1 )
352: CALL ZTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
353: IF( INFO.GT.0 )
354: + INFO = INFO + K
355: IF( INFO.GT.0 )
356: + RETURN
357: CALL ZTRMM( 'L', 'U', 'C', DIAG, K, K, CONE, A( 0 ), N+1,
358: + A( K+1 ), N+1 )
359: *
360: ELSE
361: *
362: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
363: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
364: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
365: *
366: CALL ZTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
367: IF( INFO.GT.0 )
368: + RETURN
369: CALL ZTRMM( 'L', 'L', 'C', DIAG, K, K, -CONE, A( K+1 ),
370: + N+1, A( 0 ), N+1 )
371: CALL ZTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
372: IF( INFO.GT.0 )
373: + INFO = INFO + K
374: IF( INFO.GT.0 )
375: + RETURN
376: CALL ZTRMM( 'R', 'U', 'N', DIAG, K, K, CONE, A( K ), N+1,
377: + A( 0 ), N+1 )
378: END IF
379: ELSE
380: *
381: * N is even and TRANSR = 'C'
382: *
383: IF( LOWER ) THEN
384: *
385: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
386: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
387: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
388: *
389: CALL ZTRTRI( 'U', DIAG, K, A( K ), K, INFO )
390: IF( INFO.GT.0 )
391: + RETURN
392: CALL ZTRMM( 'L', 'U', 'N', DIAG, K, K, -CONE, A( K ), K,
393: + A( K*( K+1 ) ), K )
394: CALL ZTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
395: IF( INFO.GT.0 )
396: + INFO = INFO + K
397: IF( INFO.GT.0 )
398: + RETURN
399: CALL ZTRMM( 'R', 'L', 'C', DIAG, K, K, CONE, A( 0 ), K,
400: + A( K*( K+1 ) ), K )
401: ELSE
402: *
403: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
404: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
405: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
406: *
407: CALL ZTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
408: IF( INFO.GT.0 )
409: + RETURN
410: CALL ZTRMM( 'R', 'U', 'C', DIAG, K, K, -CONE,
411: + A( K*( K+1 ) ), K, A( 0 ), K )
412: CALL ZTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
413: IF( INFO.GT.0 )
414: + INFO = INFO + K
415: IF( INFO.GT.0 )
416: + RETURN
417: CALL ZTRMM( 'L', 'L', 'N', DIAG, K, K, CONE, A( K*K ), K,
418: + A( 0 ), K )
419: END IF
420: END IF
421: END IF
422: *
423: RETURN
424: *
425: * End of ZTFTRI
426: *
427: END
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