LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ cpftrs()

 subroutine cpftrs ( character TRANSR, character UPLO, integer N, integer NRHS, complex, dimension( 0: * ) A, complex, dimension( ldb, * ) B, integer LDB, integer INFO )

CPFTRS

Purpose:
``` CPFTRS solves a system of linear equations A*X = B with a Hermitian
positive definite matrix A using the Cholesky factorization
A = U**H*U or A = L*L**H computed by CPFTRF.```
Parameters
 [in] TRANSR ``` TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugate-transpose TRANSR of RFP A is stored.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of RFP A is stored; = 'L': Lower triangle of RFP A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in] A ``` A is COMPLEX array, dimension ( N*(N+1)/2 ); The triangular factor U or L from the Cholesky factorization of RFP A = U**H*U or RFP A = L*L**H, as computed by CPFTRF. See note below for more details about RFP A.``` [in,out] B ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  We first consider Standard Packed Format when N is even.
We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N even and TRANSR = 'N'.

RFP A                   RFP A

-- -- --
03 04 05                33 43 53
-- --
13 14 15                00 44 54
--
23 24 25                10 11 55

33 34 35                20 21 22
--
00 44 45                30 31 32
-- --
01 11 55                40 41 42
-- -- --
02 12 22                50 51 52

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- -- --                -- -- -- -- -- --
03 13 23 33 00 01 02    33 00 10 20 30 40 50
-- -- -- -- --                -- -- -- -- --
04 14 24 34 44 11 12    43 44 11 21 31 41 51
-- -- -- -- -- --                -- -- -- --
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We next  consider Standard Packed Format when N is odd.
We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two   columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two   columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N odd  and TRANSR = 'N'.

RFP A                   RFP A

-- --
02 03 04                00 33 43
--
12 13 14                10 11 44

22 23 24                20 21 22
--
00 33 34                30 31 32
-- --
01 11 44                40 41 42

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- --                   -- -- -- -- -- --
02 12 22 00 01             00 10 20 30 40 50
-- -- -- --                   -- -- -- -- --
03 13 23 33 11             33 11 21 31 41 51
-- -- -- -- --                   -- -- -- --
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 219 of file cpftrs.f.

220*
221* -- LAPACK computational routine --
222* -- LAPACK is a software package provided by Univ. of Tennessee, --
223* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
224*
225* .. Scalar Arguments ..
226 CHARACTER TRANSR, UPLO
227 INTEGER INFO, LDB, N, NRHS
228* ..
229* .. Array Arguments ..
230 COMPLEX A( 0: * ), B( LDB, * )
231* ..
232*
233* =====================================================================
234*
235* .. Parameters ..
236 COMPLEX CONE
237 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
238* ..
239* .. Local Scalars ..
240 LOGICAL LOWER, NORMALTRANSR
241* ..
242* .. External Functions ..
243 LOGICAL LSAME
244 EXTERNAL lsame
245* ..
246* .. External Subroutines ..
247 EXTERNAL xerbla, ctfsm
248* ..
249* .. Intrinsic Functions ..
250 INTRINSIC max
251* ..
252* .. Executable Statements ..
253*
254* Test the input parameters.
255*
256 info = 0
257 normaltransr = lsame( transr, 'N' )
258 lower = lsame( uplo, 'L' )
259 IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'C' ) ) THEN
260 info = -1
261 ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
262 info = -2
263 ELSE IF( n.LT.0 ) THEN
264 info = -3
265 ELSE IF( nrhs.LT.0 ) THEN
266 info = -4
267 ELSE IF( ldb.LT.max( 1, n ) ) THEN
268 info = -7
269 END IF
270 IF( info.NE.0 ) THEN
271 CALL xerbla( 'CPFTRS', -info )
272 RETURN
273 END IF
274*
275* Quick return if possible
276*
277 IF( n.EQ.0 .OR. nrhs.EQ.0 )
278 \$ RETURN
279*
280* start execution: there are two triangular solves
281*
282 IF( lower ) THEN
283 CALL ctfsm( transr, 'L', uplo, 'N', 'N', n, nrhs, cone, a, b,
284 \$ ldb )
285 CALL ctfsm( transr, 'L', uplo, 'C', 'N', n, nrhs, cone, a, b,
286 \$ ldb )
287 ELSE
288 CALL ctfsm( transr, 'L', uplo, 'C', 'N', n, nrhs, cone, a, b,
289 \$ ldb )
290 CALL ctfsm( transr, 'L', uplo, 'N', 'N', n, nrhs, cone, a, b,
291 \$ ldb )
292 END IF
293*
294 RETURN
295*
296* End of CPFTRS
297*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ctfsm(TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
CTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
Definition: ctfsm.f:298
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