Uses extra precise iterative refinement to compute the solution to the system of linear equations with a banded coefficient matrix A and multiple right-hand sides
The routine uses the LU factorization to compute the solution to a real or complex system of linear equations
nbanded matrix, the columns of the matrix
Bare individual right-hand sides, and the columns of
Xare the corresponding solutions.
Both normwise and maximum componentwise error bounds are also provided on request. The routine returns a solution with a small guaranteed error (
epsis the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also calculated and returned.
The routine accepts user-provided factorizations and equilibration factors; see definitions of the
equedoptions. Solving with refinement and using a factorization from a previous call of the routine also produces a solution with
O(eps)errors or warnings but that may not be true for general user-provided factorizations and equilibration factors if they differ from what the routine would itself produce.
?gbsvxxperforms the following steps:
- If, scaling factorsfact='E'randcare computed to equilibrate the system::trans='N'diag(r)*A*diag(c)*inv(diag(c))*X=diag(r)*B:trans='T'(diag(r)*A*diag(c))*inv(Tdiag(r))*X=diag(c)*B:trans='C'(diag(r)*A*diag(c))*inv(Hdiag(r))*X=diag(c)*BWhether or not the system will be equilibrated depends on the scaling of the matrixA, but if equilibration is used,Ais overwritten byanddiag(r)*A*diag(c)Bby(ifdiag(r)*Bortrans='N')(ifdiag(c)*Bortrans='T').'C'
- Iforfact='N', the'E'LUdecomposition is used to factor the matrixA(after equilibration if) asfact='E', whereA=P*L*UPis a permutation matrix,Lis a unit lower triangular matrix, andUis upper triangular.
- If some= 0, so thatUi,iUis exactly singular, then the routine returns with. Otherwise, the factored form ofinfo=iAis used to estimate the condition number of the matrixA(see thercondparameter). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve forXand compute error bounds.
- The system of equations is solved forXusing the factored form ofA.
- By default, unlessparamsis set to zero, the routine applies iterative refinement to improve the computed solution matrix and calculate error bounds. Refinement calculates the residual to at least twice the working precision.
- If equilibration was used, the matrixXis premultiplied by(ifdiag(c)) ortrans='N'(ifdiag(r)ortrans='T') so that it solves the original system before equilibration.'C'
- Specifies whether two-dimensional array storage is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
- Must be'F','N', or'E'.Specifies whether or not the factored form of the matrixAis supplied on entry, and if not, whether the matrixAshould be equilibrated before it is factored.If, on entry,fact='F'afbandipivcontain the factored form ofA. Ifequedis not'N', the matrixAhas been equilibrated with scaling factors given byrandc. Parametersab,afb, andipivare not modified.If, the matrixfact='N'Awill be copied toafband factored.If, the matrixfact='E'Awill be equilibrated, if necessary, copied toafband factored.
- Must be'N','T', or'C'.Specifies the form of the system of equations:If, the system has the formtrans='N'(No transpose).A*X=BIf, the system has the formtrans='T'AT*X=B(Transpose).If, the system has the formtrans='C'AH*X=B(Conjugate Transpose = Transpose for real flavors, Conjugate Transpose for complex flavors).
- The number of linear equations; the order of the matrixA;n≥0.
- The number of subdiagonals within the band ofA;kl≥0.
- The number of superdiagonals within the band ofA;ku≥0.
- The number of right-hand sides; the number of columns of the matricesBandX;nrhs≥0.
- Arrays:ab(max(,ldab*n))afb(max(,ldafb*n))b(max(1,.ldb*nrhs) for column major layout and max(1,ldb*n) for row major layout)The arrayabcontains the matrixAin band storage.Ifandfact='F'equedis not'N', thenABmust have been equilibrated by the scaling factors inrand/orc.The arrayafbis an input argument if. It contains the factored form of the banded matrixfact='F'A, that is, the factorsLandUfrom the factorizationas computed byA=P*L*U?gbtrf.Uis stored as an upper triangular banded matrix withkl+kusuperdiagonals.Lis stored as lower triangular band matrix withklsubdiagonals. Ifequedis not'N', thenafbis the factored form of the equilibrated matrixA.The arraybcontains the matrixBwhose columns are the right-hand sides for the systems of equations.
- The leading dimension of the arrayab;.ldab≥kl+ku+1
- The leading dimension of the arrayafb;.ldafb≥2*kl+ku+1
- Array, size at leastmax(1,. The arrayn)ipivis an input argument if. It contains the pivot indices from the factorizationfact='F'as computed byA=P*L*U?gbtrf; rowiof the matrix was interchanged with row.ipiv[i-1]
- Must be'N','R','C', or'B'.equedis an input argument if. It specifies the form of equilibration that was done:fact='F'If, no equilibration was done (always true ifequed='N').fact='N'If, row equilibration was done, that is,equed='R'Ahas been premultiplied bydiag(r).If, column equilibration was done, that is,equed='C'Ahas been postmultiplied bydiag(c).If, both row and column equilibration was done, that is,equed='B'Ahas been replaced by.diag(r)*A*diag(c)
- Arrays:r(sizen),c(sizen). The arrayrcontains the row scale factors forA, and the arrayccontains the column scale factors forA. These arrays are input arguments ifonly; otherwise they are output arguments.fact='F'Iforequed='R''B',Ais multiplied on the left bydiag(r); iforequed='N''C',ris not accessed.Ifandfact='F'orequed='R''B', each element ofrmust be positive.Iforequed='C''B',Ais multiplied on the right bydiag(c); iforequed='N''R',cis not accessed.Ifandfact='F'orequed='C''B', each element ofcmust be positive.Each element ofrorcshould be a