001:       SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO )
002: *
003: *  -- LAPACK routine (version 3.2.1)                                    --
004: *
005: *  -- Contributed by Fred Gustavson of the IBM Watson Research Center --
006: *  -- April 2009                                                      --
007: *
008: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
009: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
010: *
011: *     .. Scalar Arguments ..
012:       CHARACTER          TRANSR, UPLO
013:       INTEGER            INFO, N
014: *     .. Array Arguments ..
015:       DOUBLE PRECISION         A( 0: * )
016: *     ..
017: *
018: *  Purpose
019: *  =======
020: *
021: *  DPFTRI computes the inverse of a (real) symmetric positive definite
022: *  matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
023: *  computed by DPFTRF.
024: *
025: *  Arguments
026: *  =========
027: *
028: *  TRANSR    (input) CHARACTER
029: *          = 'N':  The Normal TRANSR of RFP A is stored;
030: *          = 'T':  The Transpose TRANSR of RFP A is stored.
031: *
032: *  UPLO    (input) CHARACTER
033: *          = 'U':  Upper triangle of A is stored;
034: *          = 'L':  Lower triangle of A is stored.
035: *
036: *  N       (input) INTEGER
037: *          The order of the matrix A.  N >= 0.
038: *
039: *  A       (input/output) DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
040: *          On entry, the symmetric matrix A in RFP format. RFP format is
041: *          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
042: *          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
043: *          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
044: *          the transpose of RFP A as defined when
045: *          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
046: *          follows: If UPLO = 'U' the RFP A contains the nt elements of
047: *          upper packed A. If UPLO = 'L' the RFP A contains the elements
048: *          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
049: *          'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
050: *          is odd. See the Note below for more details.
051: *
052: *          On exit, the symmetric inverse of the original matrix, in the
053: *          same storage format.
054: *
055: *  INFO    (output) INTEGER
056: *          = 0:  successful exit
057: *          < 0:  if INFO = -i, the i-th argument had an illegal value
058: *          > 0:  if INFO = i, the (i,i) element of the factor U or L is
059: *                zero, and the inverse could not be computed.
060: *
061: *  Further Details
062: *  ===============
063: *
064: *  We first consider Rectangular Full Packed (RFP) Format when N is
065: *  even. We give an example where N = 6.
066: *
067: *      AP is Upper             AP is Lower
068: *
069: *   00 01 02 03 04 05       00
070: *      11 12 13 14 15       10 11
071: *         22 23 24 25       20 21 22
072: *            33 34 35       30 31 32 33
073: *               44 45       40 41 42 43 44
074: *                  55       50 51 52 53 54 55
075: *
076: *
077: *  Let TRANSR = 'N'. RFP holds AP as follows:
078: *  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
079: *  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
080: *  the transpose of the first three columns of AP upper.
081: *  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
082: *  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
083: *  the transpose of the last three columns of AP lower.
084: *  This covers the case N even and TRANSR = 'N'.
085: *
086: *         RFP A                   RFP A
087: *
088: *        03 04 05                33 43 53
089: *        13 14 15                00 44 54
090: *        23 24 25                10 11 55
091: *        33 34 35                20 21 22
092: *        00 44 45                30 31 32
093: *        01 11 55                40 41 42
094: *        02 12 22                50 51 52
095: *
096: *  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
097: *  transpose of RFP A above. One therefore gets:
098: *
099: *
100: *           RFP A                   RFP A
101: *
102: *     03 13 23 33 00 01 02    33 00 10 20 30 40 50
103: *     04 14 24 34 44 11 12    43 44 11 21 31 41 51
104: *     05 15 25 35 45 55 22    53 54 55 22 32 42 52
105: *
106: *
107: *  We first consider Rectangular Full Packed (RFP) Format when N is
108: *  odd. We give an example where N = 5.
109: *
110: *     AP is Upper                 AP is Lower
111: *
112: *   00 01 02 03 04              00
113: *      11 12 13 14              10 11
114: *         22 23 24              20 21 22
115: *            33 34              30 31 32 33
116: *               44              40 41 42 43 44
117: *
118: *
119: *  Let TRANSR = 'N'. RFP holds AP as follows:
120: *  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
121: *  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
122: *  the transpose of the first two columns of AP upper.
123: *  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
124: *  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
125: *  the transpose of the last two columns of AP lower.
126: *  This covers the case N odd and TRANSR = 'N'.
127: *
128: *         RFP A                   RFP A
129: *
130: *        02 03 04                00 33 43
131: *        12 13 14                10 11 44
132: *        22 23 24                20 21 22
133: *        00 33 34                30 31 32
134: *        01 11 44                40 41 42
135: *
136: *  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
137: *  transpose of RFP A above. One therefore gets:
138: *
139: *           RFP A                   RFP A
140: *
141: *     02 12 22 00 01             00 10 20 30 40 50
142: *     03 13 23 33 11             33 11 21 31 41 51
143: *     04 14 24 34 44             43 44 22 32 42 52
144: *
145: *  =====================================================================
146: *
147: *     .. Parameters ..
148:       DOUBLE PRECISION   ONE
149:       PARAMETER          ( ONE = 1.0D+0 )
150: *     ..
151: *     .. Local Scalars ..
152:       LOGICAL            LOWER, NISODD, NORMALTRANSR
153:       INTEGER            N1, N2, K
154: *     ..
155: *     .. External Functions ..
156:       LOGICAL            LSAME
157:       EXTERNAL           LSAME
158: *     ..
159: *     .. External Subroutines ..
160:       EXTERNAL           XERBLA, DTFTRI, DLAUUM, DTRMM, DSYRK
161: *     ..
162: *     .. Intrinsic Functions ..
163:       INTRINSIC          MOD
164: *     ..
165: *     .. Executable Statements ..
166: *
167: *     Test the input parameters.
168: *
169:       INFO = 0
170:       NORMALTRANSR = LSAME( TRANSR, 'N' )
171:       LOWER = LSAME( UPLO, 'L' )
172:       IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'T' ) ) THEN
173:          INFO = -1
174:       ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
175:          INFO = -2
176:       ELSE IF( N.LT.0 ) THEN
177:          INFO = -3
178:       END IF
179:       IF( INFO.NE.0 ) THEN
180:          CALL XERBLA( 'DPFTRI', -INFO )
181:          RETURN
182:       END IF
183: *
184: *     Quick return if possible
185: *
186:       IF( N.EQ.0 )
187:      +   RETURN
188: *
189: *     Invert the triangular Cholesky factor U or L.
190: *
191:       CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
192:       IF( INFO.GT.0 )
193:      +   RETURN
194: *
195: *     If N is odd, set NISODD = .TRUE.
196: *     If N is even, set K = N/2 and NISODD = .FALSE.
197: *
198:       IF( MOD( N, 2 ).EQ.0 ) THEN
199:          K = N / 2
200:          NISODD = .FALSE.
201:       ELSE
202:          NISODD = .TRUE.
203:       END IF
204: *
205: *     Set N1 and N2 depending on LOWER
206: *
207:       IF( LOWER ) THEN
208:          N2 = N / 2
209:          N1 = N - N2
210:       ELSE
211:          N1 = N / 2
212:          N2 = N - N1
213:       END IF
214: *
215: *     Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
216: *     inv(L)^C*inv(L). There are eight cases.
217: *
218:       IF( NISODD ) THEN
219: *
220: *        N is odd
221: *
222:          IF( NORMALTRANSR ) THEN
223: *
224: *           N is odd and TRANSR = 'N'
225: *
226:             IF( LOWER ) THEN
227: *
228: *              SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
229: *              T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
230: *              T1 -> a(0), T2 -> a(n), S -> a(N1)
231: *
232:                CALL DLAUUM( 'L', N1, A( 0 ), N, INFO )
233:                CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
234:      +                     A( 0 ), N )
235:                CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N,
236:      +                     A( N1 ), N )
237:                CALL DLAUUM( 'U', N2, A( N ), N, INFO )
238: *
239:             ELSE
240: *
241: *              SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
242: *              T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
243: *              T1 -> a(N2), T2 -> a(N1), S -> a(0)
244: *
245:                CALL DLAUUM( 'L', N1, A( N2 ), N, INFO )
246:                CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
247:      +                     A( N2 ), N )
248:                CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N,
249:      +                     A( 0 ), N )
250:                CALL DLAUUM( 'U', N2, A( N1 ), N, INFO )
251: *
252:             END IF
253: *
254:          ELSE
255: *
256: *           N is odd and TRANSR = 'T'
257: *
258:             IF( LOWER ) THEN
259: *
260: *              SRPA for LOWER, TRANSPOSE, and N is odd
261: *              T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
262: *
263:                CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO )
264:                CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
265:      +                     A( 0 ), N1 )
266:                CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1,
267:      +                     A( N1*N1 ), N1 )
268:                CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO )
269: *
270:             ELSE
271: *
272: *              SRPA for UPPER, TRANSPOSE, and N is odd
273: *              T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
274: *
275:                CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
276:                CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
277:      +                     A( N2*N2 ), N2 )
278:                CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ),
279:      +                     N2, A( 0 ), N2 )
280:                CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
281: *
282:             END IF
283: *
284:          END IF
285: *
286:       ELSE
287: *
288: *        N is even
289: *
290:          IF( NORMALTRANSR ) THEN
291: *
292: *           N is even and TRANSR = 'N'
293: *
294:             IF( LOWER ) THEN
295: *
296: *              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
297: *              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
298: *              T1 -> a(1), T2 -> a(0), S -> a(k+1)
299: *
300:                CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO )
301:                CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
302:      +                     A( 1 ), N+1 )
303:                CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1,
304:      +                     A( K+1 ), N+1 )
305:                CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO )
306: *
307:             ELSE
308: *
309: *              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
310: *              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
311: *              T1 -> a(k+1), T2 -> a(k), S -> a(0)
312: *
313:                CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO )
314:                CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
315:      +                     A( K+1 ), N+1 )
316:                CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1,
317:      +                     A( 0 ), N+1 )
318:                CALL DLAUUM( 'U', K, A( K ), N+1, INFO )
319: *
320:             END IF
321: *
322:          ELSE
323: *
324: *           N is even and TRANSR = 'T'
325: *
326:             IF( LOWER ) THEN
327: *
328: *              SRPA for LOWER, TRANSPOSE, and N is even (see paper)
329: *              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
330: *              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
331: *
332:                CALL DLAUUM( 'U', K, A( K ), K, INFO )
333:                CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
334:      +                     A( K ), K )
335:                CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
336:      +                     A( K*( K+1 ) ), K )
337:                CALL DLAUUM( 'L', K, A( 0 ), K, INFO )
338: *
339:             ELSE
340: *
341: *              SRPA for UPPER, TRANSPOSE, and N is even (see paper)
342: *              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0),
343: *              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
344: *
345:                CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
346:                CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
347:      +                     A( K*( K+1 ) ), K )
348:                CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K,
349:      +                     A( 0 ), K )
350:                CALL DLAUUM( 'L', K, A( K*K ), K, INFO )
351: *
352:             END IF
353: *
354:          END IF
355: *
356:       END IF
357: *
358:       RETURN
359: *
360: *     End of DPFTRI
361: *
362:       END
363: