 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zpftrs()

 subroutine zpftrs ( character TRANSR, character UPLO, integer N, integer NRHS, complex*16, dimension( 0: * ) A, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO )

ZPFTRS

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Purpose:
``` ZPFTRS 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 ZPFTRF.```
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*16 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 ZPFTRF. See note below for more details about RFP A.``` [in,out] B ``` B is COMPLEX*16 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```
Date
December 2016
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 222 of file zpftrs.f.

222 *
223 * -- LAPACK computational routine (version 3.7.0) --
224 * -- LAPACK is a software package provided by Univ. of Tennessee, --
225 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
226 * December 2016
227 *
228 * .. Scalar Arguments ..
229  CHARACTER transr, uplo
230  INTEGER info, ldb, n, nrhs
231 * ..
232 * .. Array Arguments ..
233  COMPLEX*16 a( 0: * ), b( ldb, * )
234 * ..
235 *
236 * =====================================================================
237 *
238 * .. Parameters ..
239  COMPLEX*16 cone
240  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
241 * ..
242 * .. Local Scalars ..
243  LOGICAL lower, normaltransr
244 * ..
245 * .. External Functions ..
246  LOGICAL lsame
247  EXTERNAL lsame
248 * ..
249 * .. External Subroutines ..
250  EXTERNAL xerbla, ztfsm
251 * ..
252 * .. Intrinsic Functions ..
253  INTRINSIC max
254 * ..
255 * .. Executable Statements ..
256 *
257 * Test the input parameters.
258 *
259  info = 0
260  normaltransr = lsame( transr, 'N' )
261  lower = lsame( uplo, 'L' )
262  IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'C' ) ) THEN
263  info = -1
264  ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
265  info = -2
266  ELSE IF( n.LT.0 ) THEN
267  info = -3
268  ELSE IF( nrhs.LT.0 ) THEN
269  info = -4
270  ELSE IF( ldb.LT.max( 1, n ) ) THEN
271  info = -7
272  END IF
273  IF( info.NE.0 ) THEN
274  CALL xerbla( 'ZPFTRS', -info )
275  RETURN
276  END IF
277 *
278 * Quick return if possible
279 *
280  IF( n.EQ.0 .OR. nrhs.EQ.0 )
281  \$ RETURN
282 *
283 * start execution: there are two triangular solves
284 *
285  IF( lower ) THEN
286  CALL ztfsm( transr, 'L', uplo, 'N', 'N', n, nrhs, cone, a, b,
287  \$ ldb )
288  CALL ztfsm( transr, 'L', uplo, 'C', 'N', n, nrhs, cone, a, b,
289  \$ ldb )
290  ELSE
291  CALL ztfsm( transr, 'L', uplo, 'C', 'N', n, nrhs, cone, a, b,
292  \$ ldb )
293  CALL ztfsm( transr, 'L', uplo, 'N', 'N', n, nrhs, cone, a, b,
294  \$ ldb )
295  END IF
296 *
297  RETURN
298 *
299 * End of ZPFTRS
300 *
subroutine ztfsm(TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
Definition: ztfsm.f:300
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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