NAG Library Routine Document F08QVF (ZTRSYL)

Transcription

1 F8 Least-squares and Eigenvalue Problems (LAPAK) F8QVF NAG Library Routine Document F8QVF (ZTRSYL) Note: before using this routine, please read the Users Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. Purpose F8QVF (ZTRSYL) solves the complex triangular Sylvester matrix equation. 2 Specification SUBROUTINE F8QVF (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB,, LD, SAL, INFO) & INTEGER ISGN, M, N, LDA, LDB, LD, INFO REAL (KIND=nag_wp) SAL OMPLEX (KIND=nag_wp) A(LDA,*), B(LDB,*), (LD,*) HARATER() TRANA, TRANB The routine may be called by its LAPAK name ztrsyl. 3 Description F8QVF (ZTRSYL) solves the complex Sylvester matrix equation opðaþx X opðbþ ¼, where opðaþ ¼ A or A H, and the matrices A and B are upper triangular; is a scale factor ( ) determined by the routine to avoid overflow in X; A is m by m and B is n by n while the right-hand side matrix and the solution matrix X are both m by n. The matrix X is obtained by a straightforward process of back-substitution (see Golub and Van Loan (996)). Note that the equation has a unique solution if and only if i j 6¼, where f i g and j are the eigenvalues of A and B respectively and the sign (þ or ) is the same as that used in the equation to be solved. 4 References Golub G H and Van Loan F (996) Matrix omputations (3rd Edition) Johns Hopkins University Press, Baltimore Higham N J (992) Perturbation theory and backward error for AX XB ¼ Numerical Analysis Report University of Manchester 5 Parameters : TRANA HARATER() Input On entry: specifies the option opðaþ. TRANA ¼ N opðaþ ¼ A. TRANA ¼ opðaþ ¼ A H. onstraint: TRANA ¼ N or. F8QVF.

2 F8QVF NAG Library Manual 2: TRANB HARATER() Input On entry: specifies the option opðbþ. TRANB ¼ N opðbþ ¼ B. TRANB ¼ opðbþ ¼ B H. onstraint: TRANB ¼ N or. 3: ISGN INTEGER Input On entry: indicates the form of the Sylvester equation. ISGN ¼þ The equation is of the form opðaþx þ X opðbþ ¼. ISGN ¼ The equation is of the form opðaþx X opðbþ ¼. onstraint: ISGN ¼þor. 4: M INTEGER Input On entry: m, the order of the matrix A, and the number of rows in the matrices X and. onstraint: M. 5: N INTEGER Input On entry: n, the order of the matrix B, and the number of columns in the matrices X and. onstraint: N. 6: AðLDA,Þ OMPLEX (KIND=nag_wp) array Input Note: the second dimension of the array A must be at least maxð; MÞ. On entry: the m by m upper triangular matrix A. 7: LDA INTEGER Input On entry: the first dimension of the array A as declared in the (sub)program from which F8QVF onstraint: LDA maxð; MÞ. 8: BðLDB,Þ OMPLEX (KIND=nag_wp) array Input Note: the second dimension of the array B must be at least maxð; NÞ. On entry: the n by n upper triangular matrix B. 9: LDB INTEGER Input On entry: the first dimension of the array B as declared in the (sub)program from which F8QVF onstraint: LDB maxð; NÞ. : ðld,þ OMPLEX (KIND=nag_wp) array Input/Output Note: the second dimension of the array must be at least maxð; NÞ. On entry: the m by n right-hand side matrix. On exit: is overwritten by the solution matrix X. F8QVF.2

3 F8 Least-squares and Eigenvalue Problems (LAPAK) F8QVF : LD INTEGER Input On entry: the first dimension of the array as declared in the (sub)program from which F8QVF onstraint: LD maxð; MÞ. 2: SAL REAL (KIND=nag_wp) Output On exit: the value of the scale factor. 3: INFO INTEGER Output On exit: INFO ¼ unless the routine detects an error (see Section 6). 6 Error Indicators and Warnings INFO < If INFO ¼ i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated. INFO ¼ A and B have common or close eigenvalues, perturbed values of which were used to solve the equation. 7 Accuracy onsider the equation AX XB ¼. replace B by B.) Let ~X be the computed solution and R the residual matrix: R ¼ A ~X ~XB. (To apply the remarks to the equation AX þ XB ¼, simply Then the residual is always small: krk F ¼ OðÞ kak F þ kbk ~ F X F. However, ~X is not necessarily the exact solution of a slightly perturbed equation; in other words, the solution is not backwards stable. For the forward error, the following bound holds: ~X X krk F F sepða; BÞ but this may be a considerable over estimate. See Golub and Van Loan (996) for a definition of sepða; BÞ, and Higham (992) for further details. These remarks also apply to the solution of a general Sylvester equation, as described in Section 8. 8 Further omments The total number of real floating point operations is approximately 4mnðm þ nþ. To solve the general complex Sylvester equation AX XB ¼ where A and B are general matrices, A and B must first be reduced to Schur form (by calling F8PNF (ZGEES), for example): A ¼ Q ~ AQ H and B ¼ Q 2 ~BQ H 2 F8QVF.3

4 F8QVF NAG Library Manual where A ~ and ~B are upper triangular and Q and Q 2 are unitary. transformed to: ~A ~X ~X ~B ¼ ~ The original equation may then be where ~X ¼ Q H XQ 2 and ~ ¼ Q H Q 2. ~ may be computed by matrix multiplication; F8QVF (ZTRSYL) may be used to solve the transformed equation; and the solution to the original equation can be obtained as X ¼ Q ~XQ H 2. The real analogue of this routine is F8QHF (DTRSYL). 9 Example This example solves the Sylvester equation AX þ XB ¼, where 6: 7:i :36 :36i :9 þ :48i :88 :25i : þ :i 5: þ 2:i :3 :72i :23 þ :3i A ¼ : þ :i : þ :i 8: :i :94 þ :53i A, : þ :i : þ :i : þ :i 3: 4:i and :5 :2i :29 :6i :37 þ :84i :55 þ :73i : þ :i :4 þ :9i :6 þ :22i :43 þ :7i B ¼ : þ :i : þ :i :9 :i :89 :42i A : þ :i : þ :i : þ :i :3 :7i :63 þ :35i :45 :56i :8 :4i :7 :23i :7 þ :9i :7 :3i :27 :54i :35 þ :2i ¼ :93 :44i :33 :35i :4 :3i :57 þ :84i A. :54 þ :25i :62 :5i :52 :3i : :8i 9. Program Text Program f8qvfe! F8QVF Example Program Text! Release. NAG opyright 22.!.. Use Statements.. Use nag_library, Only: nag_wp, x4dbf, ztrsyl!.. Implicit None Statement.. Implicit None!.. Parameters.. Integer, Parameter :: nin = 5, nout = 6!.. Local Scalars.. Real (Kind=nag_wp) :: scale Integer :: i, ifail, info, lda, ldb, ldc, m, n!.. Local Arrays.. omplex (Kind=nag_wp), Allocatable :: a(:,:), b(:,:), c(:,:) haracter () :: clabs(), rlabs()!.. Executable Statements.. Write (nout,*) F8QVF Example Program Results Write (nout,*) Flush (nout)! Skip heading in data file Read (nin,*) Read (nin,*) m, n lda = m ldb = n ldc = m Allocate (a(lda,m),b(ldb,n),c(ldc,n))! Read A, B and from data file F8QVF.4

5 F8 Least-squares and Eigenvalue Problems (LAPAK) F8QVF Read (nin,*)(a(i,:m),i=,m) Read (nin,*)(b(i,:n),i=,n) Read (nin,*)(c(i,:n),i=,m)! Solve the Sylvester equation A*X + X*B = for X! The NAG name equivalent of ztrsyl is f8qvf all ztrsyl( No transpose, No transpose,,m,n,a,lda,b,ldb,c,ldc,scale, & info) If (info>=) Then Write (nout,99999) Write (nout,*) Flush (nout) End If! Print X! ifail: behaviour on error exit! = for hard exit, = for quiet-soft, =- for noisy-soft ifail = all x4dbf( General,,m,n,c,ldc, Bracketed, F8.4, & Solution Matrix, I,rlabs, I,clabs,8,,ifail) Format (/ A and B have common or very close eigenvalues. / Pe, & rturbed values were used to solve the equations ) End Program f8qvfe 9.2 Program Data F8QVF Example Program Data 4 4 :Values of M and N (-6.,-7.) (.36,-.36) (-.9,.48) (.88,-.25) (.,.) (-5., 2.) (-.3,-.72) (-.23,.3) (.,.) (.,.) ( 8.,-.) (.94,.53) (.,.) (.,.) (.,.) ( 3.,-4.) :End of matrix A (.5,-.2) (-.29,-.6) (-.37,.84) (-.55,.73) (.,.) (-.4,.9) (.6,.22) (-.43,.7) (.,.) (.,.) (-.9,-.) (-.89,-.42) (.,.) (.,.) (.,.) (.3,-.7) :End of matrix B (.63,.35) (.45,-.56) (.8,-.4) (-.7,-.23) (-.7,.9) (-.7,-.3) (.27,-.54) (.35,.2) (-.93,-.44) (-.33,-.35) (.4,-.3) (.57,.84) (.54,.25) (-.62,-.5) (-.52,-.3) (.,-.8) :End of matrix 9.3 Program Results F8QVF Example Program Results Solution Matrix 2 3 ( -.6,.249) ( -.3,.798) ( -.62,.65) 2 (.25, -.3) ( -.55,.57) ( -.665,.78) 3 ( -.949, -.785) ( -.45, -.298) (.357,.244) 4 (.28,.52) ( -.97, -.24) ( -.27, -.94) 4 (.54, -.63) 2 (.29, ) 3 (.284,.8) 4 (.42,.48) F8QVF.5 (last)