 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ csytrs_3()

 subroutine csytrs_3 ( character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO )

CSYTRS_3

Purpose:
``` CSYTRS_3 solves a system of linear equations A * X = B with a complex
symmetric matrix A using the factorization computed
by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This algorithm is using Level 3 BLAS.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).``` [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 (LDA,N) Diagonal of the block diagonal matrix D and factors U or L as computed by CSYTRF_RK and CSYTRF_BK: a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D should be provided on entry in array E), and b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] E ``` E is COMPLEX array, dimension (N) On entry, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced; If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced. NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CSYTRF_RK or CSYTRF_BK.``` [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```
Contributors:
```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 163 of file csytrs_3.f.

165*
166* -- LAPACK computational routine --
167* -- LAPACK is a software package provided by Univ. of Tennessee, --
168* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169*
170* .. Scalar Arguments ..
171 CHARACTER UPLO
172 INTEGER INFO, LDA, LDB, N, NRHS
173* ..
174* .. Array Arguments ..
175 INTEGER IPIV( * )
176 COMPLEX A( LDA, * ), B( LDB, * ), E( * )
177* ..
178*
179* =====================================================================
180*
181* .. Parameters ..
182 COMPLEX ONE
183 parameter( one = ( 1.0e+0,0.0e+0 ) )
184* ..
185* .. Local Scalars ..
186 LOGICAL UPPER
187 INTEGER I, J, K, KP
188 COMPLEX AK, AKM1, AKM1K, BK, BKM1, DENOM
189* ..
190* .. External Functions ..
191 LOGICAL LSAME
192 EXTERNAL lsame
193* ..
194* .. External Subroutines ..
195 EXTERNAL cscal, cswap, ctrsm, xerbla
196* ..
197* .. Intrinsic Functions ..
198 INTRINSIC abs, max
199* ..
200* .. Executable Statements ..
201*
202 info = 0
203 upper = lsame( uplo, 'U' )
204 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
205 info = -1
206 ELSE IF( n.LT.0 ) THEN
207 info = -2
208 ELSE IF( nrhs.LT.0 ) THEN
209 info = -3
210 ELSE IF( lda.LT.max( 1, n ) ) THEN
211 info = -5
212 ELSE IF( ldb.LT.max( 1, n ) ) THEN
213 info = -9
214 END IF
215 IF( info.NE.0 ) THEN
216 CALL xerbla( 'CSYTRS_3', -info )
217 RETURN
218 END IF
219*
220* Quick return if possible
221*
222 IF( n.EQ.0 .OR. nrhs.EQ.0 )
223 \$ RETURN
224*
225 IF( upper ) THEN
226*
227* Begin Upper
228*
229* Solve A*X = B, where A = U*D*U**T.
230*
231* P**T * B
232*
233* Interchange rows K and IPIV(K) of matrix B in the same order
234* that the formation order of IPIV(I) vector for Upper case.
235*
236* (We can do the simple loop over IPIV with decrement -1,
237* since the ABS value of IPIV(I) represents the row index
238* of the interchange with row i in both 1x1 and 2x2 pivot cases)
239*
240 DO k = n, 1, -1
241 kp = abs( ipiv( k ) )
242 IF( kp.NE.k ) THEN
243 CALL cswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
244 END IF
245 END DO
246*
247* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
248*
249 CALL ctrsm( 'L', 'U', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
250*
251* Compute D \ B -> B [ D \ (U \P**T * B) ]
252*
253 i = n
254 DO WHILE ( i.GE.1 )
255 IF( ipiv( i ).GT.0 ) THEN
256 CALL cscal( nrhs, one / a( i, i ), b( i, 1 ), ldb )
257 ELSE IF ( i.GT.1 ) THEN
258 akm1k = e( i )
259 akm1 = a( i-1, i-1 ) / akm1k
260 ak = a( i, i ) / akm1k
261 denom = akm1*ak - one
262 DO j = 1, nrhs
263 bkm1 = b( i-1, j ) / akm1k
264 bk = b( i, j ) / akm1k
265 b( i-1, j ) = ( ak*bkm1-bk ) / denom
266 b( i, j ) = ( akm1*bk-bkm1 ) / denom
267 END DO
268 i = i - 1
269 END IF
270 i = i - 1
271 END DO
272*
273* Compute (U**T \ B) -> B [ U**T \ (D \ (U \P**T * B) ) ]
274*
275 CALL ctrsm( 'L', 'U', 'T', 'U', n, nrhs, one, a, lda, b, ldb )
276*
277* P * B [ P * (U**T \ (D \ (U \P**T * B) )) ]
278*
279* Interchange rows K and IPIV(K) of matrix B in reverse order
280* from the formation order of IPIV(I) vector for Upper case.
281*
282* (We can do the simple loop over IPIV with increment 1,
283* since the ABS value of IPIV( I ) represents the row index
284* of the interchange with row i in both 1x1 and 2x2 pivot cases)
285*
286 DO k = 1, n, 1
287 kp = abs( ipiv( k ) )
288 IF( kp.NE.k ) THEN
289 CALL cswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
290 END IF
291 END DO
292*
293 ELSE
294*
295* Begin Lower
296*
297* Solve A*X = B, where A = L*D*L**T.
298*
299* P**T * B
300* Interchange rows K and IPIV(K) of matrix B in the same order
301* that the formation order of IPIV(I) vector for Lower case.
302*
303* (We can do the simple loop over IPIV with increment 1,
304* since the ABS value of IPIV(I) represents the row index
305* of the interchange with row i in both 1x1 and 2x2 pivot cases)
306*
307 DO k = 1, n, 1
308 kp = abs( ipiv( k ) )
309 IF( kp.NE.k ) THEN
310 CALL cswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
311 END IF
312 END DO
313*
314* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
315*
316 CALL ctrsm( 'L', 'L', 'N', 'U', n, nrhs, one, a, lda, b, ldb )
317*
318* Compute D \ B -> B [ D \ (L \P**T * B) ]
319*
320 i = 1
321 DO WHILE ( i.LE.n )
322 IF( ipiv( i ).GT.0 ) THEN
323 CALL cscal( nrhs, one / a( i, i ), b( i, 1 ), ldb )
324 ELSE IF( i.LT.n ) THEN
325 akm1k = e( i )
326 akm1 = a( i, i ) / akm1k
327 ak = a( i+1, i+1 ) / akm1k
328 denom = akm1*ak - one
329 DO j = 1, nrhs
330 bkm1 = b( i, j ) / akm1k
331 bk = b( i+1, j ) / akm1k
332 b( i, j ) = ( ak*bkm1-bk ) / denom
333 b( i+1, j ) = ( akm1*bk-bkm1 ) / denom
334 END DO
335 i = i + 1
336 END IF
337 i = i + 1
338 END DO
339*
340* Compute (L**T \ B) -> B [ L**T \ (D \ (L \P**T * B) ) ]
341*
342 CALL ctrsm('L', 'L', 'T', 'U', n, nrhs, one, a, lda, b, ldb )
343*
344* P * B [ P * (L**T \ (D \ (L \P**T * B) )) ]
345*
346* Interchange rows K and IPIV(K) of matrix B in reverse order
347* from the formation order of IPIV(I) vector for Lower case.
348*
349* (We can do the simple loop over IPIV with decrement -1,
350* since the ABS value of IPIV(I) represents the row index
351* of the interchange with row i in both 1x1 and 2x2 pivot cases)
352*
353 DO k = n, 1, -1
354 kp = abs( ipiv( k ) )
355 IF( kp.NE.k ) THEN
356 CALL cswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
357 END IF
358 END DO
359*
360* END Lower
361*
362 END IF
363*
364 RETURN
365*
366* End of CSYTRS_3
367*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:180
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