1: *> \brief \b DPFTRI
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPFTRI + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpftri.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER TRANSR, UPLO
25: * INTEGER INFO, N
26: * .. Array Arguments ..
27: * DOUBLE PRECISION A( 0: * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> DPFTRI computes the inverse of a (real) symmetric positive definite
37: *> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
38: *> computed by DPFTRF.
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: *> = 'T': The 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 DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
67: *> On entry, the symmetric 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 = 'T' then RFP is
71: *> the 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: *> 'T'. 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 symmetric 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 doubleOTHERcomputational
101: *
102: *> \par Further Details:
103: * =====================
104: *>
105: *> \verbatim
106: *>
107: *> We first consider Rectangular Full Packed (RFP) Format when N is
108: *> even. 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: *> the 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: *> the transpose of the last three columns of AP lower.
127: *> This covers the case N even and TRANSR = 'N'.
128: *>
129: *> RFP A RFP A
130: *>
131: *> 03 04 05 33 43 53
132: *> 13 14 15 00 44 54
133: *> 23 24 25 10 11 55
134: *> 33 34 35 20 21 22
135: *> 00 44 45 30 31 32
136: *> 01 11 55 40 41 42
137: *> 02 12 22 50 51 52
138: *>
139: *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
140: *> transpose of RFP A above. One therefore gets:
141: *>
142: *>
143: *> RFP A RFP A
144: *>
145: *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
146: *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
147: *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
148: *>
149: *>
150: *> We then consider Rectangular Full Packed (RFP) Format when N is
151: *> odd. We give an example where N = 5.
152: *>
153: *> AP is Upper AP is Lower
154: *>
155: *> 00 01 02 03 04 00
156: *> 11 12 13 14 10 11
157: *> 22 23 24 20 21 22
158: *> 33 34 30 31 32 33
159: *> 44 40 41 42 43 44
160: *>
161: *>
162: *> Let TRANSR = 'N'. RFP holds AP as follows:
163: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
164: *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
165: *> the transpose of the first two columns of AP upper.
166: *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
167: *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
168: *> the transpose of the last two columns of AP lower.
169: *> This covers the case N odd and TRANSR = 'N'.
170: *>
171: *> RFP A RFP A
172: *>
173: *> 02 03 04 00 33 43
174: *> 12 13 14 10 11 44
175: *> 22 23 24 20 21 22
176: *> 00 33 34 30 31 32
177: *> 01 11 44 40 41 42
178: *>
179: *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
180: *> transpose of RFP A above. One therefore gets:
181: *>
182: *> RFP A RFP A
183: *>
184: *> 02 12 22 00 01 00 10 20 30 40 50
185: *> 03 13 23 33 11 33 11 21 31 41 51
186: *> 04 14 24 34 44 43 44 22 32 42 52
187: *> \endverbatim
188: *>
189: * =====================================================================
190: SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
191: *
192: * -- LAPACK computational routine --
193: * -- LAPACK is a software package provided by Univ. of Tennessee, --
194: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195: *
196: * .. Scalar Arguments ..
197: CHARACTER TRANSR, UPLO
198: INTEGER INFO, N
199: * .. Array Arguments ..
200: DOUBLE PRECISION A( 0: * )
201: * ..
202: *
203: * =====================================================================
204: *
205: * .. Parameters ..
206: DOUBLE PRECISION ONE
207: PARAMETER ( ONE = 1.0D+0 )
208: * ..
209: * .. Local Scalars ..
210: LOGICAL LOWER, NISODD, NORMALTRANSR
211: INTEGER N1, N2, K
212: * ..
213: * .. External Functions ..
214: LOGICAL LSAME
215: EXTERNAL LSAME
216: * ..
217: * .. External Subroutines ..
218: EXTERNAL XERBLA, DTFTRI, DLAUUM, DTRMM, DSYRK
219: * ..
220: * .. Intrinsic Functions ..
221: INTRINSIC MOD
222: * ..
223: * .. Executable Statements ..
224: *
225: * Test the input parameters.
226: *
227: INFO = 0
228: NORMALTRANSR = LSAME( TRANSR, 'N' )
229: LOWER = LSAME( UPLO, 'L' )
230: IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
231: INFO = -1
232: ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
233: INFO = -2
234: ELSE IF( N.LT.0 ) THEN
235: INFO = -3
236: END IF
237: IF( INFO.NE.0 ) THEN
238: CALL XERBLA( 'DPFTRI', -INFO )
239: RETURN
240: END IF
241: *
242: * Quick return if possible
243: *
244: IF( N.EQ.0 )
245: $ RETURN
246: *
247: * Invert the triangular Cholesky factor U or L.
248: *
249: CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
250: IF( INFO.GT.0 )
251: $ RETURN
252: *
253: * If N is odd, set NISODD = .TRUE.
254: * If N is even, set K = N/2 and NISODD = .FALSE.
255: *
256: IF( MOD( N, 2 ).EQ.0 ) THEN
257: K = N / 2
258: NISODD = .FALSE.
259: ELSE
260: NISODD = .TRUE.
261: END IF
262: *
263: * Set N1 and N2 depending on LOWER
264: *
265: IF( LOWER ) THEN
266: N2 = N / 2
267: N1 = N - N2
268: ELSE
269: N1 = N / 2
270: N2 = N - N1
271: END IF
272: *
273: * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
274: * inv(L)^C*inv(L). There are eight cases.
275: *
276: IF( NISODD ) THEN
277: *
278: * N is odd
279: *
280: IF( NORMALTRANSR ) THEN
281: *
282: * N is odd and TRANSR = 'N'
283: *
284: IF( LOWER ) THEN
285: *
286: * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
287: * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
288: * T1 -> a(0), T2 -> a(n), S -> a(N1)
289: *
290: CALL DLAUUM( 'L', N1, A( 0 ), N, INFO )
291: CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
292: $ A( 0 ), N )
293: CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N,
294: $ A( N1 ), N )
295: CALL DLAUUM( 'U', N2, A( N ), N, INFO )
296: *
297: ELSE
298: *
299: * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
300: * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
301: * T1 -> a(N2), T2 -> a(N1), S -> a(0)
302: *
303: CALL DLAUUM( 'L', N1, A( N2 ), N, INFO )
304: CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
305: $ A( N2 ), N )
306: CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N,
307: $ A( 0 ), N )
308: CALL DLAUUM( 'U', N2, A( N1 ), N, INFO )
309: *
310: END IF
311: *
312: ELSE
313: *
314: * N is odd and TRANSR = 'T'
315: *
316: IF( LOWER ) THEN
317: *
318: * SRPA for LOWER, TRANSPOSE, and N is odd
319: * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
320: *
321: CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO )
322: CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
323: $ A( 0 ), N1 )
324: CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1,
325: $ A( N1*N1 ), N1 )
326: CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO )
327: *
328: ELSE
329: *
330: * SRPA for UPPER, TRANSPOSE, and N is odd
331: * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
332: *
333: CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
334: CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
335: $ A( N2*N2 ), N2 )
336: CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ),
337: $ N2, A( 0 ), N2 )
338: CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
339: *
340: END IF
341: *
342: END IF
343: *
344: ELSE
345: *
346: * N is even
347: *
348: IF( NORMALTRANSR ) THEN
349: *
350: * N is even and TRANSR = 'N'
351: *
352: IF( LOWER ) THEN
353: *
354: * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
355: * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
356: * T1 -> a(1), T2 -> a(0), S -> a(k+1)
357: *
358: CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO )
359: CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
360: $ A( 1 ), N+1 )
361: CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1,
362: $ A( K+1 ), N+1 )
363: CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO )
364: *
365: ELSE
366: *
367: * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
368: * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
369: * T1 -> a(k+1), T2 -> a(k), S -> a(0)
370: *
371: CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO )
372: CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
373: $ A( K+1 ), N+1 )
374: CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1,
375: $ A( 0 ), N+1 )
376: CALL DLAUUM( 'U', K, A( K ), N+1, INFO )
377: *
378: END IF
379: *
380: ELSE
381: *
382: * N is even and TRANSR = 'T'
383: *
384: IF( LOWER ) THEN
385: *
386: * SRPA for LOWER, TRANSPOSE, and N is even (see paper)
387: * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
388: * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
389: *
390: CALL DLAUUM( 'U', K, A( K ), K, INFO )
391: CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
392: $ A( K ), K )
393: CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
394: $ A( K*( K+1 ) ), K )
395: CALL DLAUUM( 'L', K, A( 0 ), K, INFO )
396: *
397: ELSE
398: *
399: * SRPA for UPPER, TRANSPOSE, and N is even (see paper)
400: * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
401: * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
402: *
403: CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
404: CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
405: $ A( K*( K+1 ) ), K )
406: CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K,
407: $ A( 0 ), K )
408: CALL DLAUUM( 'L', K, A( K*K ), K, INFO )
409: *
410: END IF
411: *
412: END IF
413: *
414: END IF
415: *
416: RETURN
417: *
418: * End of DPFTRI
419: *
420: END
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