1: SUBROUTINE DPFTRI( TRANSR, UPLO, 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
13: INTEGER INFO, N
14: * .. Array Arguments ..
15: DOUBLE PRECISION A( 0: * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * DPFTRI computes the inverse of a (real) symmetric positive definite
22: * matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
23: * computed by DPFTRF.
24: *
25: * Arguments
26: * =========
27: *
28: * TRANSR (input) CHARACTER
29: * = 'N': The Normal TRANSR of RFP A is stored;
30: * = 'T': The Transpose TRANSR of RFP A is stored.
31: *
32: * UPLO (input) CHARACTER
33: * = 'U': Upper triangle of A is stored;
34: * = 'L': Lower triangle of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * A (input/output) DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
40: * On entry, the symmetric matrix A in RFP format. RFP format is
41: * described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
42: * then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
43: * (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
44: * the transpose of RFP A as defined when
45: * TRANSR = 'N'. The contents of RFP A are defined by UPLO as
46: * follows: If UPLO = 'U' the RFP A contains the nt elements of
47: * upper packed A. If UPLO = 'L' the RFP A contains the elements
48: * of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
49: * 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
50: * is odd. See the Note below for more details.
51: *
52: * On exit, the symmetric inverse of the original matrix, in the
53: * same storage format.
54: *
55: * INFO (output) INTEGER
56: * = 0: successful exit
57: * < 0: if INFO = -i, the i-th argument had an illegal value
58: * > 0: if INFO = i, the (i,i) element of the factor U or L is
59: * zero, and the inverse could not be computed.
60: *
61: * Further Details
62: * ===============
63: *
64: * We first consider Rectangular Full Packed (RFP) Format when N is
65: * even. We give an example where N = 6.
66: *
67: * AP is Upper AP is Lower
68: *
69: * 00 01 02 03 04 05 00
70: * 11 12 13 14 15 10 11
71: * 22 23 24 25 20 21 22
72: * 33 34 35 30 31 32 33
73: * 44 45 40 41 42 43 44
74: * 55 50 51 52 53 54 55
75: *
76: *
77: * Let TRANSR = 'N'. RFP holds AP as follows:
78: * For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
79: * three columns of AP upper. The lower triangle A(4:6,0:2) consists of
80: * the transpose of the first three columns of AP upper.
81: * For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
82: * three columns of AP lower. The upper triangle A(0:2,0:2) consists of
83: * the transpose of the last three columns of AP lower.
84: * This covers the case N even and TRANSR = 'N'.
85: *
86: * RFP A RFP A
87: *
88: * 03 04 05 33 43 53
89: * 13 14 15 00 44 54
90: * 23 24 25 10 11 55
91: * 33 34 35 20 21 22
92: * 00 44 45 30 31 32
93: * 01 11 55 40 41 42
94: * 02 12 22 50 51 52
95: *
96: * Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
97: * transpose of RFP A above. One therefore gets:
98: *
99: *
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 then 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
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