LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
spftrf.f
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1 *> \brief \b SPFTRF
2 *
3 * =========== DOCUMENTATION ===========
4 *
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6 * http://www.netlib.org/lapack/explore-html/
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPFTRF( TRANSR, UPLO, N, A, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER TRANSR, UPLO
25 * INTEGER N, INFO
26 * ..
27 * .. Array Arguments ..
28 * REAL A( 0: * )
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> SPFTRF computes the Cholesky factorization of a real symmetric
37 *> positive definite matrix A.
38 *>
39 *> The factorization has the form
40 *> A = U**T * U, if UPLO = 'U', or
41 *> A = L * L**T, if UPLO = 'L',
42 *> where U is an upper triangular matrix and L is lower triangular.
43 *>
44 *> This is the block version of the algorithm, calling Level 3 BLAS.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] TRANSR
51 *> \verbatim
52 *> TRANSR is CHARACTER*1
53 *> = 'N': The Normal TRANSR of RFP A is stored;
54 *> = 'T': The Transpose TRANSR of RFP A is stored.
55 *> \endverbatim
56 *>
57 *> \param[in] UPLO
58 *> \verbatim
59 *> UPLO is CHARACTER*1
60 *> = 'U': Upper triangle of RFP A is stored;
61 *> = 'L': Lower triangle of RFP A is stored.
62 *> \endverbatim
63 *>
64 *> \param[in] N
65 *> \verbatim
66 *> N is INTEGER
67 *> The order of the matrix A. N >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in,out] A
71 *> \verbatim
72 *> A is REAL array, dimension ( N*(N+1)/2 );
73 *> On entry, the symmetric matrix A in RFP format. RFP format is
74 *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
75 *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
76 *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
77 *> the transpose of RFP A as defined when
78 *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as
79 *> follows: If UPLO = 'U' the RFP A contains the NT elements of
80 *> upper packed A. If UPLO = 'L' the RFP A contains the elements
81 *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
82 *> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
83 *> is odd. See the Note below for more details.
84 *>
85 *> On exit, if INFO = 0, the factor U or L from the Cholesky
86 *> factorization RFP A = U**T*U or RFP A = L*L**T.
87 *> \endverbatim
88 *>
89 *> \param[out] INFO
90 *> \verbatim
91 *> INFO is INTEGER
92 *> = 0: successful exit
93 *> < 0: if INFO = -i, the i-th argument had an illegal value
94 *> > 0: if INFO = i, the leading minor of order i is not
95 *> positive definite, and the factorization could not be
96 *> completed.
97 *> \endverbatim
98 *
99 * Authors:
100 * ========
101 *
102 *> \author Univ. of Tennessee
103 *> \author Univ. of California Berkeley
104 *> \author Univ. of Colorado Denver
105 *> \author NAG Ltd.
106 *
107 *> \date November 2011
108 *
109 *> \ingroup realOTHERcomputational
110 *
111 *> \par Further Details:
112 * =====================
113 *>
114 *> \verbatim
115 *>
116 *> We first consider Rectangular Full Packed (RFP) Format when N is
117 *> even. We give an example where N = 6.
118 *>
119 *> AP is Upper AP is Lower
120 *>
121 *> 00 01 02 03 04 05 00
122 *> 11 12 13 14 15 10 11
123 *> 22 23 24 25 20 21 22
124 *> 33 34 35 30 31 32 33
125 *> 44 45 40 41 42 43 44
126 *> 55 50 51 52 53 54 55
127 *>
128 *>
129 *> Let TRANSR = 'N'. RFP holds AP as follows:
130 *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
131 *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of
132 *> the transpose of the first three columns of AP upper.
133 *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
134 *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of
135 *> the transpose of the last three columns of AP lower.
136 *> This covers the case N even and TRANSR = 'N'.
137 *>
138 *> RFP A RFP A
139 *>
140 *> 03 04 05 33 43 53
141 *> 13 14 15 00 44 54
142 *> 23 24 25 10 11 55
143 *> 33 34 35 20 21 22
144 *> 00 44 45 30 31 32
145 *> 01 11 55 40 41 42
146 *> 02 12 22 50 51 52
147 *>
148 *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
149 *> transpose of RFP A above. One therefore gets:
150 *>
151 *>
152 *> RFP A RFP A
153 *>
154 *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
155 *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
156 *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52
157 *>
158 *>
159 *> We then consider Rectangular Full Packed (RFP) Format when N is
160 *> odd. We give an example where N = 5.
161 *>
162 *> AP is Upper AP is Lower
163 *>
164 *> 00 01 02 03 04 00
165 *> 11 12 13 14 10 11
166 *> 22 23 24 20 21 22
167 *> 33 34 30 31 32 33
168 *> 44 40 41 42 43 44
169 *>
170 *>
171 *> Let TRANSR = 'N'. RFP holds AP as follows:
172 *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
173 *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of
174 *> the transpose of the first two columns of AP upper.
175 *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
176 *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of
177 *> the transpose of the last two columns of AP lower.
178 *> This covers the case N odd and TRANSR = 'N'.
179 *>
180 *> RFP A RFP A
181 *>
182 *> 02 03 04 00 33 43
183 *> 12 13 14 10 11 44
184 *> 22 23 24 20 21 22
185 *> 00 33 34 30 31 32
186 *> 01 11 44 40 41 42
187 *>
188 *> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
189 *> transpose of RFP A above. One therefore gets:
190 *>
191 *> RFP A RFP A
192 *>
193 *> 02 12 22 00 01 00 10 20 30 40 50
194 *> 03 13 23 33 11 33 11 21 31 41 51
195 *> 04 14 24 34 44 43 44 22 32 42 52
196 *> \endverbatim
197 *>
198 * =====================================================================
199  SUBROUTINE spftrf( TRANSR, UPLO, N, A, INFO )
200 *
201 * -- LAPACK computational routine (version 3.4.0) --
202 * -- LAPACK is a software package provided by Univ. of Tennessee, --
203 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
204 * November 2011
205 *
206 * .. Scalar Arguments ..
207  CHARACTER TRANSR, UPLO
208  INTEGER N, INFO
209 * ..
210 * .. Array Arguments ..
211  REAL A( 0: * )
212 *
213 * =====================================================================
214 *
215 * .. Parameters ..
216  REAL ONE
217  parameter ( one = 1.0e+0 )
218 * ..
219 * .. Local Scalars ..
220  LOGICAL LOWER, NISODD, NORMALTRANSR
221  INTEGER N1, N2, K
222 * ..
223 * .. External Functions ..
224  LOGICAL LSAME
225  EXTERNAL lsame
226 * ..
227 * .. External Subroutines ..
228  EXTERNAL xerbla, ssyrk, spotrf, strsm
229 * ..
230 * .. Intrinsic Functions ..
231  INTRINSIC mod
232 * ..
233 * .. Executable Statements ..
234 *
235 * Test the input parameters.
236 *
237  info = 0
238  normaltransr = lsame( transr, 'N' )
239  lower = lsame( uplo, 'L' )
240  IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'T' ) ) THEN
241  info = -1
242  ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
243  info = -2
244  ELSE IF( n.LT.0 ) THEN
245  info = -3
246  END IF
247  IF( info.NE.0 ) THEN
248  CALL xerbla( 'SPFTRF', -info )
249  RETURN
250  END IF
251 *
252 * Quick return if possible
253 *
254  IF( n.EQ.0 )
255  \$ RETURN
256 *
257 * If N is odd, set NISODD = .TRUE.
258 * If N is even, set K = N/2 and NISODD = .FALSE.
259 *
260  IF( mod( n, 2 ).EQ.0 ) THEN
261  k = n / 2
262  nisodd = .false.
263  ELSE
264  nisodd = .true.
265  END IF
266 *
267 * Set N1 and N2 depending on LOWER
268 *
269  IF( lower ) THEN
270  n2 = n / 2
271  n1 = n - n2
272  ELSE
273  n1 = n / 2
274  n2 = n - n1
275  END IF
276 *
277 * start execution: there are eight cases
278 *
279  IF( nisodd ) THEN
280 *
281 * N is odd
282 *
283  IF( normaltransr ) THEN
284 *
285 * N is odd and TRANSR = 'N'
286 *
287  IF( lower ) THEN
288 *
289 * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:n1-1) )
290 * T1 -> a(0,0), T2 -> a(0,1), S -> a(n1,0)
291 * T1 -> a(0), T2 -> a(n), S -> a(n1)
292 *
293  CALL spotrf( 'L', n1, a( 0 ), n, info )
294  IF( info.GT.0 )
295  \$ RETURN
296  CALL strsm( 'R', 'L', 'T', 'N', n2, n1, one, a( 0 ), n,
297  \$ a( n1 ), n )
298  CALL ssyrk( 'U', 'N', n2, n1, -one, a( n1 ), n, one,
299  \$ a( n ), n )
300  CALL spotrf( 'U', n2, a( n ), n, info )
301  IF( info.GT.0 )
302  \$ info = info + n1
303 *
304  ELSE
305 *
306 * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:n2-1)
307 * T1 -> a(n1+1,0), T2 -> a(n1,0), S -> a(0,0)
308 * T1 -> a(n2), T2 -> a(n1), S -> a(0)
309 *
310  CALL spotrf( 'L', n1, a( n2 ), n, info )
311  IF( info.GT.0 )
312  \$ RETURN
313  CALL strsm( 'L', 'L', 'N', 'N', n1, n2, one, a( n2 ), n,
314  \$ a( 0 ), n )
315  CALL ssyrk( 'U', 'T', n2, n1, -one, a( 0 ), n, one,
316  \$ a( n1 ), n )
317  CALL spotrf( 'U', n2, a( n1 ), n, info )
318  IF( info.GT.0 )
319  \$ info = info + n1
320 *
321  END IF
322 *
323  ELSE
324 *
325 * N is odd and TRANSR = 'T'
326 *
327  IF( lower ) THEN
328 *
329 * SRPA for LOWER, TRANSPOSE and N is odd
330 * T1 -> A(0,0) , T2 -> A(1,0) , S -> A(0,n1)
331 * T1 -> a(0+0) , T2 -> a(1+0) , S -> a(0+n1*n1); lda=n1
332 *
333  CALL spotrf( 'U', n1, a( 0 ), n1, info )
334  IF( info.GT.0 )
335  \$ RETURN
336  CALL strsm( 'L', 'U', 'T', 'N', n1, n2, one, a( 0 ), n1,
337  \$ a( n1*n1 ), n1 )
338  CALL ssyrk( 'L', 'T', n2, n1, -one, a( n1*n1 ), n1, one,
339  \$ a( 1 ), n1 )
340  CALL spotrf( 'L', n2, a( 1 ), n1, info )
341  IF( info.GT.0 )
342  \$ info = info + n1
343 *
344  ELSE
345 *
346 * SRPA for UPPER, TRANSPOSE and N is odd
347 * T1 -> A(0,n1+1), T2 -> A(0,n1), S -> A(0,0)
348 * T1 -> a(n2*n2), T2 -> a(n1*n2), S -> a(0); lda = n2
349 *
350  CALL spotrf( 'U', n1, a( n2*n2 ), n2, info )
351  IF( info.GT.0 )
352  \$ RETURN
353  CALL strsm( 'R', 'U', 'N', 'N', n2, n1, one, a( n2*n2 ),
354  \$ n2, a( 0 ), n2 )
355  CALL ssyrk( 'L', 'N', n2, n1, -one, a( 0 ), n2, one,
356  \$ a( n1*n2 ), n2 )
357  CALL spotrf( 'L', n2, a( n1*n2 ), n2, info )
358  IF( info.GT.0 )
359  \$ info = info + n1
360 *
361  END IF
362 *
363  END IF
364 *
365  ELSE
366 *
367 * N is even
368 *
369  IF( normaltransr ) THEN
370 *
371 * N is even and TRANSR = 'N'
372 *
373  IF( lower ) THEN
374 *
375 * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
376 * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
377 * T1 -> a(1), T2 -> a(0), S -> a(k+1)
378 *
379  CALL spotrf( 'L', k, a( 1 ), n+1, info )
380  IF( info.GT.0 )
381  \$ RETURN
382  CALL strsm( 'R', 'L', 'T', 'N', k, k, one, a( 1 ), n+1,
383  \$ a( k+1 ), n+1 )
384  CALL ssyrk( 'U', 'N', k, k, -one, a( k+1 ), n+1, one,
385  \$ a( 0 ), n+1 )
386  CALL spotrf( 'U', k, a( 0 ), n+1, info )
387  IF( info.GT.0 )
388  \$ info = info + k
389 *
390  ELSE
391 *
392 * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
393 * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
394 * T1 -> a(k+1), T2 -> a(k), S -> a(0)
395 *
396  CALL spotrf( 'L', k, a( k+1 ), n+1, info )
397  IF( info.GT.0 )
398  \$ RETURN
399  CALL strsm( 'L', 'L', 'N', 'N', k, k, one, a( k+1 ),
400  \$ n+1, a( 0 ), n+1 )
401  CALL ssyrk( 'U', 'T', k, k, -one, a( 0 ), n+1, one,
402  \$ a( k ), n+1 )
403  CALL spotrf( 'U', k, a( k ), n+1, info )
404  IF( info.GT.0 )
405  \$ info = info + k
406 *
407  END IF
408 *
409  ELSE
410 *
411 * N is even and TRANSR = 'T'
412 *
413  IF( lower ) THEN
414 *
415 * SRPA for LOWER, TRANSPOSE and N is even (see paper)
416 * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1)
417 * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
418 *
419  CALL spotrf( 'U', k, a( 0+k ), k, info )
420  IF( info.GT.0 )
421  \$ RETURN
422  CALL strsm( 'L', 'U', 'T', 'N', k, k, one, a( k ), n1,
423  \$ a( k*( k+1 ) ), k )
424  CALL ssyrk( 'L', 'T', k, k, -one, a( k*( k+1 ) ), k, one,
425  \$ a( 0 ), k )
426  CALL spotrf( 'L', k, a( 0 ), k, info )
427  IF( info.GT.0 )
428  \$ info = info + k
429 *
430  ELSE
431 *
432 * SRPA for UPPER, TRANSPOSE and N is even (see paper)
433 * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0)
434 * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
435 *
436  CALL spotrf( 'U', k, a( k*( k+1 ) ), k, info )
437  IF( info.GT.0 )
438  \$ RETURN
439  CALL strsm( 'R', 'U', 'N', 'N', k, k, one,
440  \$ a( k*( k+1 ) ), k, a( 0 ), k )
441  CALL ssyrk( 'L', 'N', k, k, -one, a( 0 ), k, one,
442  \$ a( k*k ), k )
443  CALL spotrf( 'L', k, a( k*k ), k, info )
444  IF( info.GT.0 )
445  \$ info = info + k
446 *
447  END IF
448 *
449  END IF
450 *
451  END IF
452 *
453  RETURN
454 *
455 * End of SPFTRF
456 *
457  END
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:171
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:183
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:109
subroutine spftrf(TRANSR, UPLO, N, A, INFO)
SPFTRF
Definition: spftrf.f:200