1: SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
2: *
3: * -- LAPACK routine (version 3.2.2) --
4: *
5: * -- Contributed by Fred Gustavson of the IBM Watson Research Center --
6: * -- June 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: DOUBLE PRECISION A( 0: * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * DTFTRI 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
31: * = 'N': The Normal TRANSR of RFP A is stored;
32: * = 'T': The Transpose TRANSR of RFP A is stored.
33: *
34: * UPLO (input) CHARACTER
35: * = 'U': A is upper triangular;
36: * = 'L': A is lower triangular.
37: *
38: * DIAG (input) CHARACTER
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) DOUBLE PRECISION array, dimension (0:nt-1);
46: * nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
47: * Positive Definite 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 = 'T' then RFP is
51: * the 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 nt
55: * elements of lower packed A. The LDA of RFP A is (N+1)/2 when
56: * TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
57: * even and N is odd. See the Note below for more details.
58: *
59: * On exit, the (triangular) inverse of the original matrix, in
60: * the same storage format.
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, A(i,i) is exactly zero. The triangular
66: * matrix is singular and its inverse can not be computed.
67: *
68: * Further Details
69: * ===============
70: *
71: * We first consider Rectangular Full Packed (RFP) Format when N is
72: * even. We give an example where N = 6.
73: *
74: * AP is Upper AP is Lower
75: *
76: * 00 01 02 03 04 05 00
77: * 11 12 13 14 15 10 11
78: * 22 23 24 25 20 21 22
79: * 33 34 35 30 31 32 33
80: * 44 45 40 41 42 43 44
81: * 55 50 51 52 53 54 55
82: *
83: *
84: * Let TRANSR = 'N'. RFP holds AP as follows:
85: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
86: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
87: * the transpose of the first three columns of AP upper.
88: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
89: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
90: * the transpose of the last three columns of AP lower.
91: * This covers the case N even and TRANSR = 'N'.
92: *
93: * RFP A RFP A
94: *
95: * 03 04 05 33 43 53
96: * 13 14 15 00 44 54
97: * 23 24 25 10 11 55
98: * 33 34 35 20 21 22
99: * 00 44 45 30 31 32
100: * 01 11 55 40 41 42
101: * 02 12 22 50 51 52
102: *
103: * Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
104: * transpose of RFP A above. One therefore gets:
105: *
106: *
107: * RFP A RFP A
108: *
109: * 03 13 23 33 00 01 02 33 00 10 20 30 40 50
110: * 04 14 24 34 44 11 12 43 44 11 21 31 41 51
111: * 05 15 25 35 45 55 22 53 54 55 22 32 42 52
112: *
113: *
114: * We then consider Rectangular Full Packed (RFP) Format when N is
115: * odd. We give an example where N = 5.
116: *
117: * AP is Upper AP is Lower
118: *
119: * 00 01 02 03 04 00
120: * 11 12 13 14 10 11
121: * 22 23 24 20 21 22
122: * 33 34 30 31 32 33
123: * 44 40 41 42 43 44
124: *
125: *
126: * Let TRANSR = 'N'. RFP holds AP as follows:
127: * For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
128: * three columns of AP upper. The lower triangle A(3:4,0:1) consists of
129: * the transpose of the first two columns of AP upper.
130: * For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
131: * three columns of AP lower. The upper triangle A(0:1,1:2) consists of
132: * the transpose of the last two columns of AP lower.
133: * This covers the case N odd and TRANSR = 'N'.
134: *
135: * RFP A RFP A
136: *
137: * 02 03 04 00 33 43
138: * 12 13 14 10 11 44
139: * 22 23 24 20 21 22
140: * 00 33 34 30 31 32
141: * 01 11 44 40 41 42
142: *
143: * Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
144: * transpose of RFP A above. One therefore gets:
145: *
146: * RFP A RFP A
147: *
148: * 02 12 22 00 01 00 10 20 30 40 50
149: * 03 13 23 33 11 33 11 21 31 41 51
150: * 04 14 24 34 44 43 44 22 32 42 52
151: *
152: * =====================================================================
153: *
154: * .. Parameters ..
155: DOUBLE PRECISION ONE
156: PARAMETER ( ONE = 1.0D+0 )
157: * ..
158: * .. Local Scalars ..
159: LOGICAL LOWER, NISODD, NORMALTRANSR
160: INTEGER N1, N2, K
161: * ..
162: * .. External Functions ..
163: LOGICAL LSAME
164: EXTERNAL LSAME
165: * ..
166: * .. External Subroutines ..
167: EXTERNAL XERBLA, DTRMM, DTRTRI
168: * ..
169: * .. Intrinsic Functions ..
170: INTRINSIC MOD
171: * ..
172: * .. Executable Statements ..
173: *
174: * Test the input parameters.
175: *
176: INFO = 0
177: NORMALTRANSR = LSAME( TRANSR, 'N' )
178: LOWER = LSAME( UPLO, 'L' )
179: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
180: INFO = -1
181: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
182: INFO = -2
183: ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND. .NOT.LSAME( DIAG, 'U' ) )
184: + THEN
185: INFO = -3
186: ELSE IF( N.LT.0 ) THEN
187: INFO = -4
188: END IF
189: IF( INFO.NE.0 ) THEN
190: CALL XERBLA( 'DTFTRI', -INFO )
191: RETURN
192: END IF
193: *
194: * Quick return if possible
195: *
196: IF( N.EQ.0 )
197: + RETURN
198: *
199: * If N is odd, set NISODD = .TRUE.
200: * If N is even, set K = N/2 and NISODD = .FALSE.
201: *
202: IF( MOD( N, 2 ).EQ.0 ) THEN
203: K = N / 2
204: NISODD = .FALSE.
205: ELSE
206: NISODD = .TRUE.
207: END IF
208: *
209: * Set N1 and N2 depending on LOWER
210: *
211: IF( LOWER ) THEN
212: N2 = N / 2
213: N1 = N - N2
214: ELSE
215: N1 = N / 2
216: N2 = N - N1
217: END IF
218: *
219: *
220: * start execution: there are eight cases
221: *
222: IF( NISODD ) THEN
223: *
224: * N is odd
225: *
226: IF( NORMALTRANSR ) THEN
227: *
228: * N is odd and TRANSR = 'N'
229: *
230: IF( LOWER ) THEN
231: *
232: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
233: * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
234: * T1 -> a(0), T2 -> a(n), S -> a(n1)
235: *
236: CALL DTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO )
237: IF( INFO.GT.0 )
238: + RETURN
239: CALL DTRMM( 'R', 'L', 'N', DIAG, N2, N1, -ONE, A( 0 ),
240: + N, A( N1 ), N )
241: CALL DTRTRI( 'U', DIAG, N2, A( N ), N, INFO )
242: IF( INFO.GT.0 )
243: + INFO = INFO + N1
244: IF( INFO.GT.0 )
245: + RETURN
246: CALL DTRMM( 'L', 'U', 'T', DIAG, N2, N1, ONE, A( N ), N,
247: + A( N1 ), N )
248: *
249: ELSE
250: *
251: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
252: * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
253: * T1 -> a(n2), T2 -> a(n1), S -> a(0)
254: *
255: CALL DTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO )
256: IF( INFO.GT.0 )
257: + RETURN
258: CALL DTRMM( 'L', 'L', 'T', DIAG, N1, N2, -ONE, A( N2 ),
259: + N, A( 0 ), N )
260: CALL DTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO )
261: IF( INFO.GT.0 )
262: + INFO = INFO + N1
263: IF( INFO.GT.0 )
264: + RETURN
265: CALL DTRMM( 'R', 'U', 'N', DIAG, N1, N2, ONE, A( N1 ),
266: + N, A( 0 ), N )
267: *
268: END IF
269: *
270: ELSE
271: *
272: * N is odd and TRANSR = 'T'
273: *
274: IF( LOWER ) THEN
275: *
276: * SRPA for LOWER, TRANSPOSE and N is odd
277: * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1)
278: *
279: CALL DTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO )
280: IF( INFO.GT.0 )
281: + RETURN
282: CALL DTRMM( 'L', 'U', 'N', DIAG, N1, N2, -ONE, A( 0 ),
283: + N1, A( N1*N1 ), N1 )
284: CALL DTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO )
285: IF( INFO.GT.0 )
286: + INFO = INFO + N1
287: IF( INFO.GT.0 )
288: + RETURN
289: CALL DTRMM( 'R', 'L', 'T', DIAG, N1, N2, ONE, A( 1 ),
290: + N1, A( N1*N1 ), N1 )
291: *
292: ELSE
293: *
294: * SRPA for UPPER, TRANSPOSE and N is odd
295: * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0)
296: *
297: CALL DTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO )
298: IF( INFO.GT.0 )
299: + RETURN
300: CALL DTRMM( 'R', 'U', 'T', DIAG, N2, N1, -ONE,
301: + A( N2*N2 ), N2, A( 0 ), N2 )
302: CALL DTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO )
303: IF( INFO.GT.0 )
304: + INFO = INFO + N1
305: IF( INFO.GT.0 )
306: + RETURN
307: CALL DTRMM( 'L', 'L', 'N', DIAG, N2, N1, ONE,
308: + A( N1*N2 ), N2, A( 0 ), N2 )
309: END IF
310: *
311: END IF
312: *
313: ELSE
314: *
315: * N is even
316: *
317: IF( NORMALTRANSR ) THEN
318: *
319: * N is even and TRANSR = 'N'
320: *
321: IF( LOWER ) THEN
322: *
323: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
324: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
325: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
326: *
327: CALL DTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO )
328: IF( INFO.GT.0 )
329: + RETURN
330: CALL DTRMM( 'R', 'L', 'N', DIAG, K, K, -ONE, A( 1 ),
331: + N+1, A( K+1 ), N+1 )
332: CALL DTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO )
333: IF( INFO.GT.0 )
334: + INFO = INFO + K
335: IF( INFO.GT.0 )
336: + RETURN
337: CALL DTRMM( 'L', 'U', 'T', DIAG, K, K, ONE, A( 0 ), N+1,
338: + A( K+1 ), N+1 )
339: *
340: ELSE
341: *
342: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
343: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
344: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
345: *
346: CALL DTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO )
347: IF( INFO.GT.0 )
348: + RETURN
349: CALL DTRMM( 'L', 'L', 'T', DIAG, K, K, -ONE, A( K+1 ),
350: + N+1, A( 0 ), N+1 )
351: CALL DTRTRI( 'U', DIAG, K, A( K ), N+1, INFO )
352: IF( INFO.GT.0 )
353: + INFO = INFO + K
354: IF( INFO.GT.0 )
355: + RETURN
356: CALL DTRMM( 'R', 'U', 'N', DIAG, K, K, ONE, A( K ), N+1,
357: + A( 0 ), N+1 )
358: END IF
359: ELSE
360: *
361: * N is even and TRANSR = 'T'
362: *
363: IF( LOWER ) THEN
364: *
365: * SRPA for LOWER, TRANSPOSE and N is even (see paper)
366: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
367: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
368: *
369: CALL DTRTRI( 'U', DIAG, K, A( K ), K, INFO )
370: IF( INFO.GT.0 )
371: + RETURN
372: CALL DTRMM( 'L', 'U', 'N', DIAG, K, K, -ONE, A( K ), K,
373: + A( K*( K+1 ) ), K )
374: CALL DTRTRI( 'L', DIAG, K, A( 0 ), K, INFO )
375: IF( INFO.GT.0 )
376: + INFO = INFO + K
377: IF( INFO.GT.0 )
378: + RETURN
379: CALL DTRMM( 'R', 'L', 'T', DIAG, K, K, ONE, A( 0 ), K,
380: + A( K*( K+1 ) ), K )
381: ELSE
382: *
383: * SRPA for UPPER, TRANSPOSE and N is even (see paper)
384: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
385: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
386: *
387: CALL DTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO )
388: IF( INFO.GT.0 )
389: + RETURN
390: CALL DTRMM( 'R', 'U', 'T', DIAG, K, K, -ONE,
391: + A( K*( K+1 ) ), K, A( 0 ), K )
392: CALL DTRTRI( 'L', DIAG, K, A( K*K ), K, INFO )
393: IF( INFO.GT.0 )
394: + INFO = INFO + K
395: IF( INFO.GT.0 )
396: + RETURN
397: CALL DTRMM( 'L', 'L', 'N', DIAG, K, K, ONE, A( K*K ), K,
398: + A( 0 ), K )
399: END IF
400: END IF
401: END IF
402: *
403: RETURN
404: *
405: * End of DTFTRI
406: *
407: END
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