c \BeginDoc
c
c \Name: dnband
c
c \Description:
c
c This subroutine returns the converged approximations to eigenvalues
c of A*z = lambda*B*z and (optionally):
c
c (1) The corresponding approximate eigenvectors;
c
c (2) An orthonormal basis for the associated approximate
c invariant subspace;
c
c (3) Both.
c
c Matrices A and B are stored in LAPACK-style banded form.
c
c There is negligible additional cost to obtain eigenvectors. An orthonormal
c basis is always computed. There is an additional storage cost of n*nev
c if both are requested (in this case a separate array Z must be supplied).
c
c The approximate eigenvalues and vectors are commonly called Ritz
c values and Ritz vectors respectively. They are referred to as such
c in the comments that follow. The computed orthonormal basis for the
c invariant subspace corresponding to these Ritz values is referred to as a
c Schur basis.
c
c dnband can be called with one of the following modes:
c
c Mode 1: A*z = lambda*z.
c ===> OP = A and B = I.
c
c Mode 2: A*z = lambda*M*z, M symmetric positive definite
c ===> OP = inv[M]*A and B = M.
c
c Mode 3: A*z = lambda*M*z, M symmetric semi-definite
c ===> OP = Real_Part{ inv[A - sigma*M]*M } and B = M.
c ===> shift-and-invert mode (in real arithmetic)
c If OP*z = amu*z, then
c amu = 1/2 * [ 1/(lambda-sigma) + 1/(lambda-conjg(sigma)) ].
c Note: If sigma is real, i.e. imaginary part of sigma is zero;
c Real_Part{ inv[A - sigma*M]*M } == inv[A - sigma*M]*M
c amu == 1/(lambda-sigma).
c
c Mode 4: A*z = lambda*M*z, M symmetric semi-definite
c ===> OP = Imaginary_Part{ inv[A - sigma*M]*M } and B = M.
c ===> shift-and-invert mode (in real arithmetic)
c If OP*z = amu*z, then
c amu = 1/2i * [ 1/(lambda-sigma) - 1/(lambda-conjg(sigma)) ].
c
c
c The choice of mode must be specified in IPARAM(7) defined below.
c
c \Usage
c call dnband
c ( RVEC, HOWMNY, SELECT, DR, DI, Z, LDZ, SIGMAR, SIGMAI,
c WORKEV, V, N, AB, MB, LDA, RFAC, CFAC, KL, KU, WHICH,
c BMAT, NEV, TOL, RESID, NCV, V, LDV, IPARAM, WORKD,
c WORKL, LWORKL, WORKC, IWORK, INFO )
c
c \Arguments
c
c RVEC LOGICAL (INPUT)
c Specifies whether a basis for the invariant subspace corresponding
c to the converged Ritz value approximations for the eigenproblem
c A*z = lambda*B*z is computed.
c
c RVEC = .FALSE. Compute Ritz values only.
c
c RVEC = .TRUE. Compute the Ritz vectors or Schur vectors.
c See Remarks below.
c
c HOWMNY Character*1 (INPUT)
c Specifies the form of the basis for the invariant subspace
c corresponding to the converged Ritz values that is to be computed.
c
c = 'A': Compute NEV Ritz vectors;
c = 'P': Compute NEV Schur vectors;
c = 'S': compute some of the Ritz vectors, specified
c by the logical array SELECT.
c
c SELECT Logical array of dimension NCV. (INPUT)
c If HOWMNY = 'S', SELECT specifies the Ritz vectors to be
c computed. To select the Ritz vector corresponding to a
c Ritz value (DR(j), DI(j)), SELECT(j) must be set to .TRUE..
c If HOWMNY = 'A' or 'P', SELECT is used as internal workspace.
c
c DR Double precision array of dimension NEV+1. (OUTPUT)
c On exit, DR contains the real part of the Ritz value approximations
c to the eigenvalues of A*z = lambda*B*z.
c
c DI Double precision array of dimension NEV+1. (OUTPUT)
c On exit, DI contains the imaginary part of the Ritz value
c approximations to the eigenvalues of A*z = lambda*B*z associated
c with DR.
c
c NOTE: When Ritz values are complex, they will come in complex
c conjugate pairs. If eigenvectors are requested, the
c corresponding Ritz vectors will also come in conjugate
c pairs and the real and imaginary parts of these are
c represented in two consecutive columns of the array Z
c (see below).
c
c Z Real N by NEV+1 array if RVEC = .TRUE. and HOWMNY = 'A'. (OUTPUT)
c On exit,
c if RVEC = .TRUE. and HOWMNY = 'A', then the columns of
c Z represent approximate eigenvectors (Ritz vectors) corresponding
c to the NCONV=IPARAM(5) Ritz values for eigensystem
c A*z = lambda*B*z computed by DNAUPD.
c
c The complex Ritz vector associated with the Ritz value
c with positive imaginary part is stored in two consecutive
c columns. The first column holds the real part of the Ritz
c vector and the second column holds the imaginary part. The
c Ritz vector associated with the Ritz value with negative
c imaginary part is simply the complex conjugate of the Ritz vector
c associated with the positive imaginary part.
c
c If RVEC = .FALSE. or HOWMNY = 'P', then Z is not referenced.
c
c NOTE: If if RVEC = .TRUE. and a Schur basis is not required,
c the array Z may be set equal to first NEV+1 columns of the Arnoldi
c basis array V computed by DNAUPD. In this case the Arnoldi basis
c will be destroyed and overwritten with the eigenvector basis.
c
c LDZ Integer. (INPUT)
c The leading dimension of the array Z. If Ritz vectors are
c desired, then LDZ >= max( 1, N ). In any case, LDZ >= 1.
c
c SIGMAR Double precision (INPUT)
c If IPARAM(7) = 3 or 4, represents the real part of the shift.
c Not referenced if IPARAM(7) = 1 or 2.
c
c SIGMAI Double precision (INPUT)
c If IPARAM(7) = 3 or 4, represents the imaginary part of the
c shift.
c Not referenced if IPARAM(7) = 1 or 2.
c
c WORKEV Double precision work array of dimension 3*NCV. (WORKSPACE)
c
c N Integer. (INPUT)
c Dimension of the eigenproblem.
c
c AB Double precision array of dimension LDA by N. (INPUT)
c The matrix A in band storage, in rows KL+1 to
c 2*KL+KU+1; rows 1 to KL of the array need not be set.
c The j-th column of A is stored in the j-th column of the
c array AB as follows:
c AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
c
c MB Double precision array of dimension LDA by N. (INPUT)
c The matrix M in band storage, in rows KL+1 to
c 2*KL+KU+1; rows 1 to KL of the array need not be set.
c The j-th column of M is stored in the j-th column of the
c array AB as follows:
c MB(kl+ku+1+i-j,j) = M(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
c Not referenced if IPARAM(7) = 1
c
c LDA Integer. (INPUT)
c Leading dimension of AB, MB, RFAC and CFAC.
c
c RFAC Double precision array of LDA by N. (WORKSPACE/OUTPUT)
c RFAC is used to store the LU factors of MB when IPARAM(7) = 2
c is invoked. It is used to store the LU factors of
c (A-sigma*M) when IPARAM(7) = 3 is invoked with a real shift.
c It is not referenced when IPARAM(7) = 1 or 4.
c
c CFAC Complex*16 array of LDA by N. (WORKSPACE/OUTPUT)
c CFAC is used to store (A-SIGMA*M) and its LU factors
c when IPARAM(7) = 3 or 4 are used with a complex shift SIGMA.
c On exit, it contains the LU factors of (A-SIGMA*M).
c It is not referenced when IPARAM(7) = 1 or 2.
c
c KL Integer. (INPUT)
c Max(number of subdiagonals of A, number of subdiagonals of M)
c
c KU Integer. (OUTPUT)
c Max(number of superdiagonals of A, number of superdiagonals of M)
c
c WHICH Character*2. (INPUT)
c When IPARAM(7)= 1 or 2, WHICH can be set to any one of
c the following.
c
c 'LM' -> want the NEV eigenvalues of largest magnitude.
c 'SM' -> want the NEV eigenvalues of smallest magnitude.
c 'LR' -> want the NEV eigenvalues of largest real part.
c 'SR' -> want the NEV eigenvalues of smallest real part.
c 'LI' -> want the NEV eigenvalues of largest imaginary part.
c 'SI' -> want the NEV eigenvalues of smallest imaginary part.
c
c When IPARAM(7) = 3 or 4, WHICH should be set to 'LM' only.
c
c BMAT Character*1. (INPUT)
c BMAT specifies the type of the matrix B that defines the
c semi-inner product for the operator OP.
c BMAT = 'I' -> standard eigenvalue problem A*z = lambda*z
c BMAT = 'G' -> generalized eigenvalue problem A*z = lambda*M*z
c NEV Integer. (INPUT)
c Number of eigenvalues to be computed.
c
c TOL Double precision scalar. (INPUT)
c Stopping criteria: the relative accuracy of the Ritz value
c is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I)).
c If TOL .LE. 0. is passed a default is set:
c DEFAULT = DLAMCH('EPS') (machine precision as computed
c by the LAPACK auxiliary subroutine DLAMCH).
c
c RESID Double precision array of length N. (INPUT/OUTPUT)
c On INPUT:
c If INFO .EQ. 0, a random initial residual vector is used.
c If INFO .NE. 0, RESID contains the initial residual vector,
c possibly from a previous run.
c On OUTPUT:
c RESID contains the final residual vector.
c
c NCV Integer. (INPUT)
c Number of columns of the matrix V (less than or equal to N).
c Represents the dimension of the Arnoldi basis constructed
c by dnaupd for OP.
c
c V Double precision array N by NCV+1. (OUTPUT)
c Upon OUTPUT: If RVEC = .TRUE. the first NCONV=IPARAM(5) columns
c represent approximate Schur vectors that span the
c desired invariant subspace.
c NOTE: The array Z may be set equal to first NEV+1 columns of the
c Arnoldi basis vector array V computed by DNAUPD. In this case
c if RVEC = .TRUE. and HOWMNY='A', then the first NCONV=IPARAM(5)
c are the desired Ritz vectors.
c
c LDV Integer. (INPUT)
c Leading dimension of V exactly as declared in the calling
c program.
c
c IPARAM Integer array of length 11. (INPUT/OUTPUT)
c IPARAM(1) = ISHIFT:
c The shifts selected at each iteration are used to restart
c the Arnoldi iteration in an implicit fashion.
c It is set to 1 in this subroutine. The user do not need
c to set this parameter.
c ----------------------------------------------------------
c ISHIFT = 1: exact shift with respect to the current
c Hessenberg matrix H. This is equivalent to
c restarting the iteration from the beginning
c after updating the starting vector with a linear
c combination of Ritz vectors associated with the
c "wanted" eigenvalues.
c -------------------------------------------------------------
c
c IPARAM(2) = No longer referenced.
c
c IPARAM(3) = MXITER
c On INPUT: max number of Arnoldi update iterations allowed.
c On OUTPUT: actual number of Arnoldi update iterations taken.
c
c IPARAM(4) = NB: blocksize to be used in the recurrence.
c The code currently works only for NB = 1.
c
c IPARAM(5) = NCONV: number of "converged" eigenvalues.
c
c IPARAM(6) = IUPD
c Not referenced. Implicit restarting is ALWAYS used.
c
c IPARAM(7) = IPARAM(7):
c On INPUT determines what type of eigenproblem is being solved.
c Must be 1,2,3,4; See under \Description of dnband for the
c four modes available.
c
c IPARAM(9) = NUMOP, IPARAM(10) = NUMOPB, IPARAM(11) = NUMREO,
c OUTPUT: NUMOP = total number of OP*z operations,
c NUMOPB = total number of B*z operations if BMAT='G',
c NUMREO = total number of steps of re-orthogonalization.
c
c WORKD Double precision work array of length at least 3*n. (WORKSPACE)
c
c WORKL Double precision work array of length LWORKL. (WORKSPACE)
c
c LWORKL Integer. (INPUT)
c LWORKL must be at least 3*NCV**2 + 6*NCV.
c
c WORKC Complex*16 array of length N. (WORKSPACE)
c Workspace used when IPARAM(7) = 3 or 4 for storing a temporary
c complex vector.
c
c IWORK Integer array of dimension at least N. (WORKSPACE)
c Used when IPARAM(7)=2,3,4 to store the pivot information in the
c factorization of M or (A-SIGMA*M).
c
c INFO Integer. (INPUT/OUTPUT)
c Error flag on output.
c = 0: Normal exit.
c = 1: The Schur form computed by LAPACK routine dlahqr
c could not be reordered by LAPACK routine dtrsen.
c Re-enter subroutine DNEUPD with IPARAM(5)=NCV and
c increase the size of the arrays DR and DI to have
c dimension at least NCV and allocate at least NCV
c columns for Z. NOTE: Not necessary if Z and V share
c the same space. Please notify the authors.
c
c = -1: N must be positive.
c = -2: NEV must be positive.
c = -3: NCV-NEV >= 2 and less than or equal to N.
c = -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'
c = -6: BMAT must be one of 'I' or 'G'.
c = -7: Length of private work WORKL array is not sufficient.
c = -8: Error return from calculation of a real Schur form.
c Informational error from LAPACK routine dlahqr.
c = -9: Error return from calculation of eigenvectors.
c Informational error from LAPACK routine dtrevc.
c = -10: IPARAM(7) must be 1,2,3,4.
c = -11: IPARAM(7) = 1 and BMAT = 'G' are incompatible.
c = -12: HOWMNY = 'S' not yet implemented
c = -13: HOWMNY must be one of 'A' or 'P'
c = -14: DNAUPD did not find any eigenvalues to sufficient
c accuracy.
c = -15: Overflow occurs when we try to transform the Ritz
c values returned from DNAUPD to those of the original
c problem using Rayleigh Quotient.
c = -9999: Could not build an Arnoldi factorization.
c IPARAM(5) returns the size of the current
c Arnoldi factorization.
c
c \EndDoc
c
c------------------------------------------------------------------------
c
c\BeginLib
c
c\References:
c 1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
c a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
c pp 357-385.
c
c 2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly
c Restarted Arnoldi Iteration", Ph.D thesis, TR95-13, Rice Univ,
c May 1995.
c
c\Routines called:
c dnaupd ARPACK reverse communication interface routine.
c dneupd ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c dgbtrf LAPACK band matrix factorization routine.
c dgbtrs LAPACK band linear system solve routine.
c zgbtrf LAPACK complex band matrix factorization routine.
c zgbtrs LAPACK complex linear system solve routine.
c dlacpy LAPACK matrix copy routine.
c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c dlamch LAPACK routine to compute the underflow threshold.
c dcopy Level 1 BLAS that copies one vector to another.
c ddot Level 1 BLAS that computes the dot product of two vectors.
c dnrm2 Level 1 BLAS that computes the norm of a vector.
c dgbmv Level 2 BLAS that computes the band matrix vector product.
c
c\Remarks
c
c 1. Currently only HOWMNY = 'A' and 'P' are implemented.
c
c Let X' denote the transpose of X.
c
c 2. Schur vectors are an orthogonal representation for the basis of
c Ritz vectors. Thus, their numerical properties are often superior.
c If RVEC = .TRUE. then the relationship
c A * V(:,1:IPARAM(5)) = V(:,1:IPARAM(5)) * T, and
c V(:,1:IPARAM(5))' * V(:,1:IPARAM(5)) = I are approximately satisfied.
c Here T is the leading submatrix of order IPARAM(5) of the real
c upper quasi-triangular matrix stored workl(ipntr(12)). That is,
c T is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
c each 2-by-2 diagonal block has its diagonal elements equal and its
c off-diagonal elements of opposite sign. Corresponding to each 2-by-2
c diagonal block is a complex conjugate pair of Ritz values. The real
c Ritz values are stored on the diagonal of T.
c
c\Author
c Danny Sorensen
c Richard Lehoucq
c Chao Yang
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\SCCS Information: @(#)
c FILE: nband.F SID: 2.3 DATE OF SID: 10/17/00 RELEASE: 2
c
c\EndLib
c
c---------------------------------------------------------------------
c
subroutine dnband( rvec, howmny, select, dr, di, z, ldz, sigmar,
& sigmai, workev, n, ab, mb, lda, rfac, cfac, kl, ku,
& which, bmat, nev, tol, resid, ncv, v, ldv,
& iparam, workd, workl, lworkl, workc, iwork, info)
c
c %------------------%
c | Scalar Arguments |
c %------------------%
c
character which*2, bmat, howmny
integer n, lda, kl, ku, nev, ncv, ldv,
& ldz, lworkl, info
Double precision
& tol, sigmar, sigmai
c
c %-----------------%
c | Array Arguments |
c %-----------------%
c
integer iparam(*), iwork(*)
logical select(*)
Double precision
& dr(*), di(*), resid(*), v(ldv,*), z(ldz,*),
& ab(lda,*), mb(lda,*), rfac(lda,*),
& workd(*), workl(*), workev(*)
Complex*16
& cfac(lda,*), workc(*)
c
c %--------------%
c | Local Arrays |
c %--------------%
c
integer ipntr(14)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
integer ido, i, j, type, imid, itop, ibot, ierr
Double precision
& numr, denr, deni, dmdul, safmin
logical rvec, first
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& one, zero
parameter (one = 1.0D+0, zero = 0.0D+0)
c
c
c %-----------------------------%
c | LAPACK & BLAS routines used |
c %-----------------------------%
c
Double precision
& ddot, dnrm2, dlapy2, dlamch
external ddot, dcopy, dgbmv, zgbtrf, zgbtrs, dgbtrf,
& dgbtrs, dnrm2, dlapy2, dlacpy, dlamch
c
c %---------------------%
c | Intrinsic Functions |
c %---------------------%
c
Intrinsic dble, dimag, dcmplx
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
c %--------------------------------%
c | safmin = safe minimum is such |
c | that 1/sfmin does not overflow |
c %--------------------------------%
c
safmin = dlamch('safmin')
c
c %----------------------------------------------------------------%
c | Set type of the problem to be solved. Check consistency |
c | between BMAT and IPARAM(7). |
c | type = 1 --> Solving standard problem in regular mode. |
c | type = 2 --> Solving standard problem in shift-invert mode. |
c | type = 3 --> Solving generalized problem in regular mode. |
c | type = 4 --> Solving generalized problem in shift-invert mode. |
c | type = 5 --> Solving standard problem in shift-invert mode |
c | using iparam(7) = 4 in DNAUPD. |
c | type = 6 --> Solving generalized problem in shift-invert mode. |
c | using iparam(7) = 4 in DNAUPD. |
c %----------------------------------------------------------------%
c
if ( iparam(7) .eq. 1 ) then
type = 1
else if ( iparam(7) .eq. 3 .and. bmat .eq. 'I') then
type = 2
else if ( iparam(7) .eq. 2 ) then
type = 3
else if ( iparam(7) .eq. 3 .and. bmat .eq. 'G') then
type = 4
else if ( iparam(7) .eq. 4 .and. bmat .eq. 'I') then
type = 5
else if ( iparam(7) .eq. 4 .and. bmat .eq. 'G') then
type = 6
else
print*, ' '
print*, 'BMAT is inconsistent with IPARAM(7).'
print*, ' '
go to 9000
end if
c
c %----------------------------------%
c | When type = 5,6 are used, sigmai |
c | must be nonzero. |
c %----------------------------------%
c
if ( type .eq. 5 .or. type .eq. 6 ) then
if ( sigmai .eq. zero ) then
print*, ' '
print*, '_NBAND: sigmai must be nonzero when type 5 or 6
& is used. '
print*, ' '
go to 9000
end if
end if
c
c %------------------------%
c | Initialize the reverse |
c | communication flag. |
c %------------------------%
c
ido = 0
c
c %----------------%
c | Exact shift is |
c | used. |
c %----------------%
c
iparam(1) = 1
c
c %-----------------------------------%
c | Both matrices A and M are stored |
c | between rows itop and ibot. Imid |
c | is the index of the row that |
c | stores the diagonal elements. |
c %-----------------------------------%
c
itop = kl + 1
imid = kl + ku + 1
ibot = 2*kl + ku + 1
c
if ( type .eq. 2 .or. type .eq. 5 ) then
c
c %-------------------------------%
c | Solving a standard eigenvalue |
c | problem in shift-invert mode. |
c | Factor (A-sigma*I). |
c %-------------------------------%
c
if (sigmai .eq. zero) then
c
c %-----------------------------------%
c | Construct (A-sigmar*I) and factor |
c | in real arithmetic. |
c %-----------------------------------%
c
call dlacpy ('A', ibot, n, ab, lda, rfac, lda )
do 10 j = 1, n
rfac(imid,j) = ab(imid,j) - sigmar
10 continue
call dgbtrf(n, n, kl, ku, rfac, lda, iwork, ierr )
if (ierr .ne. 0) then
print*, ' '
print*, ' _NBAND: Error with _gbtrf. '
print*, ' '
go to 9000
end if
c
else
c
c %-----------------------------------%
c | Construct (A-sigmar*I) and factor |
c | in COMPLEX arithmetic. |
c %-----------------------------------%
c
do 30 j = 1, n
do 20 i = itop, ibot
cfac(i,j) = dcmplx(ab(i,j))
20 continue
30 continue
c
do 40 j = 1, n
cfac(imid,j) = cfac(imid,j)
$ - dcmplx(sigmar, sigmai)
40 continue
c
call zgbtrf(n, n, kl, ku, cfac, lda, iwork, ierr )
if ( ierr .ne. 0) then
print*, ' '
print*, ' _NBAND: Error with _gbtrf. '
print*, ' '
go to 9000
end if
c
end if
else if ( type .eq. 3 ) then
c
c %-----------------------------------------------%
c | Solving generalized eigenvalue problem in |
c | regular mode. Copy M to rfac, and call LAPACK |
c | routine dgbtrf to factor M. |
c %-----------------------------------------------%
c
call dlacpy ('A', ibot, n, mb, lda, rfac, lda )
call dgbtrf(n, n, kl, ku, rfac, lda, iwork, ierr)
if (ierr .ne. 0) then
print*, ' '
print*,'_NBAND: Error with _gbtrf.'
print*, ' '
go to 9000
end if
c
else if ( type .eq. 4 .or. type .eq. 6 ) then
c
c %-------------------------------------------%
c | Solving generalized eigenvalue problem in |
c | shift-invert mode. |
c %-------------------------------------------%
c
if ( sigmai .eq. zero ) then
c
c %--------------------------------------------%
c | Construct (A - sigma*M) and factor in real |
c | arithmetic. |
c %--------------------------------------------%
c
do 60 j = 1,n
do 50 i = itop, ibot
rfac(i,j) = ab(i,j) - sigmar*mb(i,j)
50 continue
60 continue
c
call dgbtrf(n, n, kl, ku, rfac, lda, iwork, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, '_NBAND: Error with _gbtrf.'
print*, ' '
go to 9000
end if
c
else
c
c %-----------------------------------------------%
c | Construct (A - sigma*M) and factor in complex |
c | arithmetic. |
c %-----------------------------------------------%
c
do 80 j = 1,n
do 70 i = itop, ibot
cfac(i,j) = dcmplx( ab(i,j)-sigmar*mb(i,j),
& -sigmai*mb(i,j) )
70 continue
80 continue
c
call zgbtrf(n, n, kl, ku, cfac, lda, iwork, ierr)
if ( ierr .NE. 0 ) then
print*, ' '
print*, '_NBAND: Error with _gbtrf.'
print*, ' '
go to 9000
end if
c
end if
c
end if
c
c %--------------------------------------------%
c | M A I N L O O P (reverse communication) |
c %--------------------------------------------%
c
90 continue
c
call dnaupd ( ido, bmat, n, which, nev, tol, resid, ncv,
& v, ldv, iparam, ipntr, workd, workl, lworkl,
& info )
c
if (ido .eq. -1) then
c
if ( type .eq. 1) then
c
c %----------------------------%
c | Perform y <--- OP*x = A*x |
c %----------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, ab(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
else if ( type .eq. 2 ) then
c
if (sigmai .eq. zero) then
c
c %----------------------------------%
c | Shift is real. Perform |
c | y <--- OP*x = inv[A-sigmar*I]*x |
c | to force the starting vector |
c | into the range of OP. |
c %----------------------------------%
c
call dcopy (n, workd(ipntr(1)), 1, workd(ipntr(2)), 1)
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' _NBAND: Error with _bgtrs. '
print*, ' '
go to 9000
end if
c
else
c
c %--------------------------------------------%
c | Shift is COMPLEX. Perform |
c | y <--- OP*x = Real_Part{inv[A-sigma*I]*x} |
c | to force the starting vector into the |
c | range of OP. |
c %--------------------------------------------%
c
do 100 j = 1, n
workc(j) = dcmplx(workd(ipntr(1)+j-1))
100 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' _NBAND: Error with _gbtrs. '
print*, ' '
go to 9000
end if
c
do 110 j = 1, n
workd(ipntr(2)+j-1) = dble(workc(j))
110 continue
c
end if
c
else if ( type .eq. 3 ) then
c
c %-----------------------------------%
c | Perform y <--- OP*x = inv[M]*A*x |
c | to force the starting vector into |
c | the range of OP. |
c %-----------------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, ab(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _bgtrs.'
print*, ' '
go to 9000
end if
c
else if ( type .eq. 4 ) then
c
c %-----------------------------------------%
c | Perform y <-- OP*x |
c | = Real_part{inv[A-SIGMA*M]*M}*x |
c | to force the starting vector into the |
c | range of OP. |
c %-----------------------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, mb(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
if ( sigmai .eq. zero ) then
c
c %---------------------%
c | Shift is real, stay |
c | in real arithmetic. |
c %---------------------%
c
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
else
c
c %--------------------------%
c | Goto complex arithmetic. |
c %--------------------------%
c
do 120 i = 1,n
workc(i) = dcmplx(workd(ipntr(2)+i-1))
120 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
do 130 i = 1, n
workd(ipntr(2)+i-1) = dble(workc(i))
130 continue
c
end if
c
else if ( type .eq. 5) then
c
c %---------------------------------------%
c | Perform y <-- OP*x |
c | = Imaginary_part{inv[A-SIGMA*I]}*x |
c | to force the starting vector into the |
c | range of OP. |
c %---------------------------------------%
c
do 140 j = 1, n
workc(j) = dcmplx(workd(ipntr(1)+j-1))
140 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' _NBAND: Error with _gbtrs. '
print*, ' '
go to 9000
end if
c
do 150 j = 1, n
workd(ipntr(2)+j-1) = dimag(workc(j))
150 continue
c
else if ( type .eq. 6 ) then
c
c %----------------------------------------%
c | Perform y <-- OP*x |
c | Imaginary_part{inv[A-SIGMA*M]*M} |
c | to force the starting vector into the |
c | range of OP. |
c %----------------------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, mb(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
do 160 i = 1,n
workc(i) = dcmplx(workd(ipntr(2)+i-1))
160 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
do 170 i = 1, n
workd(ipntr(2)+i-1) = dimag(workc(i))
170 continue
c
end if
c
else if (ido .eq. 1) then
c
if ( type .eq. 1) then
c
c %----------------------------%
c | Perform y <--- OP*x = A*x |
c %----------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, ab(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
else if ( type .eq. 2) then
c
if ( sigmai .eq. zero) then
c
c %----------------------------------%
c | Shift is real. Perform |
c | y <--- OP*x = inv[A-sigmar*I]*x. |
c %----------------------------------%
c
call dcopy (n, workd(ipntr(1)), 1, workd(ipntr(2)), 1)
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
else
c
c %------------------------------------------%
c | Shift is COMPLEX. Perform |
c | y <-- OP*x = Real_Part{inv[A-sigma*I]*x} |
c | in COMPLEX arithmetic. |
c %------------------------------------------%
c
do 180 j = 1, n
workc(j) = dcmplx(workd(ipntr(1)+j-1))
180 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
do 190 j = 1, n
workd(ipntr(2)+j-1) = dble(workc(j))
190 continue
c
end if
c
else if ( type .eq. 3 ) then
c
c %-----------------------------------%
c | Perform y <--- OP*x = inv[M]*A*x |
c %-----------------------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, ab(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _bgtrs.'
print*, ' '
go to 9000
end if
c
else if ( type .eq. 4 ) then
c
c %--------------------------------------%
c | Perform y <-- inv(A-sigma*M)*(M*x). |
c | (M*x) has been computed and stored |
c | in workd(ipntr(3)). |
c %--------------------------------------%
c
if ( sigmai .eq. zero ) then
c
c %------------------------%
c | Shift is real, stay in |
c | real arithmetic. |
c %------------------------%
c
call dcopy(n, workd(ipntr(3)), 1, workd(ipntr(2)), 1)
call dgbtrs ('Notranspose', n, kl, ku, 1, rfac, lda,
& iwork, workd(ipntr(2)), n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
else
c
c %---------------------------%
c | Go to COMPLEX arithmetic. |
c %---------------------------%
c
do 200 i = 1,n
workc(i) = dcmplx(workd(ipntr(3)+i-1))
200 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error in _gbtrs.'
print*, ' '
go to 9000
end if
c
do 210 i = 1,n
workd(ipntr(2)+i-1) = dble(workc(i))
210 continue
c
end if
c
else if ( type .eq. 5 ) then
c
c %---------------------------------------%
c | Perform y <-- OP*x |
c | = Imaginary_part{inv[A-SIGMA*I]*x} |
c %---------------------------------------%
c
do 220 j = 1, n
workc(j) = dcmplx(workd(ipntr(1)+j-1))
220 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' _NBAND: Error with _gbtrs. '
print*, ' '
go to 9000
end if
c
do 230 j = 1, n
workd(ipntr(2)+j-1) = dimag(workc(j))
230 continue
c
else if ( type .eq. 6) then
c
c %-----------------------------------------%
c | Perform y <-- OP*x |
c | = Imaginary_part{inv[A-SIGMA*M]*M}*x. |
c %-----------------------------------------%
c
do 240 i = 1,n
workc(i) = dcmplx(workd(ipntr(3)+i-1))
240 continue
c
call zgbtrs ('Notranspose', n, kl, ku, 1, cfac, lda,
& iwork, workc, n, ierr)
if (ierr .ne. 0) then
print*, ' '
print*, '_NBAND: Error with _gbtrs.'
print*, ' '
go to 9000
end if
c
do 250 i = 1, n
workd(ipntr(2)+i-1) = dimag(workc(i))
250 continue
c
end if
c
else if (ido .eq. 2) then
c
c %--------------------%
c | Perform y <-- M*x |
c | Not used when |
c | type = 1,2. |
c %--------------------%
c
call dgbmv('Notranspose', n, n, kl, ku, one, mb(itop,1),
& lda, workd(ipntr(1)), 1, zero,
& workd(ipntr(2)), 1)
c
else
c
c %-----------------------------------------%
c | Either we have convergence, or there is |
c | error. |
c %-----------------------------------------%
c
if ( info .lt. 0) then
c
c %--------------------------%
c | Error message, check the |
c | documentation in DNAUPD |
c %--------------------------%
c
print *, ' '
print *, ' Error with _naupd info = ',info
print *, ' Check the documentation of _naupd '
print *, ' '
go to 9000
c
else
c
if ( info .eq. 1) then
print *, ' '
print *, ' Maximum number of iterations reached.'
print *, ' '
else if ( info .eq. 3) then
print *, ' '
print *, ' No shifts could be applied during implicit',
& ' Arnoldi update, try increasing NCV.'
print *, ' '
end if
c
if (iparam(5) .gt. 0) then
c
call dneupd ( rvec, 'A', select, dr, di, z, ldz,
& sigmar, sigmai, workev, bmat, n, which,
& nev, tol, resid, ncv, v, ldv, iparam,
& ipntr, workd, workl, lworkl, info )
c
if ( info .ne. 0) then
c
c %------------------------------------%
c | Check the documentation of DNEUPD. |
c %------------------------------------%
c
print *, ' '
print *, ' Error with _neupd = ', info
print *, ' Check the documentation of _neupd '
print *, ' '
go to 9000
c
else if ( sigmai .ne. zero ) then
c
if ( type .eq. 4 .or. type .eq. 6 ) then
c
first = .true.
do 270 j = 1, iparam(5)
c
c %----------------------------------%
c | Use Rayleigh Quotient to recover |
c | eigenvalues of the original |
c | generalized eigenvalue problem. |
c %----------------------------------%
c
if ( di(j) .eq. zero ) then
c
c %--------------------------------------%
c | Eigenvalue is real. Compute |
c | d = (x'*inv[A-sigma*M]*M*x) / (x'*x) |
c %--------------------------------------%
c
call dgbmv('Nontranspose', n, n, kl, ku, one,
$ mb(itop,1), lda, z(1,j), 1, zero,
$ workd, 1)
do i = 1, n
workc(i) = dcmplx(workd(i))
end do
call zgbtrs ('Notranspose', n, kl, ku, 1,
$ cfac, lda, iwork, workc, n, info)
do i = 1, n
workd(i) = dble(workc(i))
workd(i+n) = dimag(workc(i))
end do
denr = ddot(n, z(1,j), 1, workd, 1)
deni = ddot(n, z(1,j), 1, workd(n+1), 1)
numr = dnrm2(n, z(1,j), 1)**2
dmdul = dlapy2(denr,deni)**2
if ( dmdul .ge. safmin ) then
dr(j) = sigmar + numr*denr / dmdul
else
c
c %---------------------%
c | dmdul is too small. |
c | Exit to avoid |
c | overflow. |
c %---------------------%
c
info = -15
go to 9000
end if
c
else if (first) then
c
c %------------------------%
c | Eigenvalue is complex. |
c | Compute the first one |
c | of the conjugate pair. |
c %------------------------%
c
c %-------------%
c | Compute M*x |
c %-------------%
c
call dgbmv('Nontranspose', n, n, kl, ku,
$ one, mb(itop,1), lda, z(1,j), 1, zero,
$ workd, 1)
call dgbmv('Nontranspose', n, n, kl, ku,
$ one, mb(itop,1), lda, z(1,j+1), 1,
$ zero, workd(n+1), 1)
do i = 1, n
workc(i) = dcmplx(workd(i),workd(i+n))
end do
c
c %----------------------------%
c | Compute inv(A-sigma*M)*M*x |
c %----------------------------%
c
call zgbtrs('Notranspose',n,kl,ku,1,cfac,
$ lda, iwork, workc, n, info)
c
c %-------------------------------%
c | Compute x'*inv(A-sigma*M)*M*x |
c %-------------------------------%
c
do i = 1, n
workd(i) = dble(workc(i))
workd(i+n) = dimag(workc(i))
end do
denr = ddot(n,z(1,j),1,workd,1)
denr = denr+ddot(n,z(1,j+1),1,workd(n+1),1)
deni = ddot(n,z(1,j),1,workd(n+1),1)
deni = deni - ddot(n,z(1,j+1),1,workd,1)
c
c %----------------%
c | Compute (x'*x) |
c %----------------%
c
numr = dlapy2( dnrm2(n, z(1,j), 1),
& dnrm2(n, z(1, j+1), 1) )**2
c
c %----------------------------------------%
c | Compute (x'x) / (x'*inv(A-sigma*M)*Mx) |
c %----------------------------------------%
c
dmdul = dlapy2(denr,deni)**2
if ( dmdul .ge. safmin ) then
dr(j) = sigmar+numr*denr / dmdul
di(j) = sigmai-numr*deni / dmdul
first = .false.
else
c
c %---------------------%
c | dmdul is too small. |
c | Exit to avoid |
c | overflow. |
c %---------------------%
c
info = -15
go to 9000
c
end if
c
else
c
c %---------------------------%
c | Get the second eigenvalue |
c | of the conjugate pair by |
c | taking the conjugate of |
c | previous one. |
c %---------------------------%
c
dr(j) = dr(j-1)
di(j) = -di(j-1)
first = .true.
c
end if
c
270 continue
c
else if ( type .eq. 2 .or. type .eq. 5) then
c
first = .true.
do 280 j = 1, iparam(5)
c
c %----------------------------------%
c | Use Rayleigh Quotient to recover |
c | eigenvalues of the original |
c | standard eigenvalue problem. |
c %----------------------------------%
c
if ( di(j) .eq. zero ) then
c
c %-------------------------------------%
c | Eigenvalue is real. Compute |
c | d = (x'*inv[A-sigma*I]*x) / (x'*x). |
c %-------------------------------------%
c
do i = 1, n
workc(i) = dcmplx(z(i,j))
end do
call zgbtrs ('Notranspose', n, kl, ku, 1,
$ cfac, lda, iwork, workc, n, info)
do i = 1, n
workd(i) = dble(workc(i))
workd(i+n) = dimag(workc(i))
end do
denr = ddot(n,z(1,j),1,workd,1)
deni = ddot(n,z(1,j),1,workd(n+1),1)
numr = dnrm2(n, z(1,j), 1)**2
dmdul = dlapy2(denr,deni)**2
if ( dmdul .ge. safmin ) then
dr(j) = sigmar + numr*denr / dmdul
else
c
c %---------------------%
c | dmdul is too small. |
c | Exit to avoid |
c | overflow. |
c %---------------------%
c
info = -15
go to 9000
c
end if
c
else if (first) then
c
c %------------------------%
c | Eigenvalue is complex. |
c | Compute the first one |
c | of the conjugate pair. |
c %------------------------%
c
do i = 1, n
workc(i) = dcmplx( z(i,j), z(i,j+1) )
end do
c
c %---------------------------%
c | Compute inv[A-sigma*I]*x. |
c %---------------------------%
c
call zgbtrs('Notranspose',n,kl,ku,1,cfac,
$ lda, iwork, workc, n, info)
c
c %-----------------------------%
c | Compute x'*inv(A-sigma*I)*x |
c %-----------------------------%
c
do i = 1, n
workd(i) = dble(workc(i))
workd(i+n) = dimag(workc(i))
end do
denr = ddot(n,z(1,j),1,workd,1)
denr = denr+ddot(n,z(1,j+1),1,workd(n+1),1)
deni = ddot(n,z(1,j),1,workd(n+1),1)
deni = deni - ddot(n,z(1,j+1),1,workd,1)
c
c %----------------%
c | Compute (x'*x) |
c %----------------%
c
numr = dlapy2( dnrm2(n, z(1,j), 1),
& dnrm2(n, z(1,j+1), 1))**2
c
c %----------------------------------------%
c | Compute (x'x) / (x'*inv(A-sigma*I)*x). |
c %----------------------------------------%
c
dmdul = dlapy2(denr,deni)**2
if (dmdul .ge. safmin) then
dr(j) = sigmar+numr*denr / dmdul
di(j) = sigmai-numr*deni / dmdul
first = .false.
else
c
c %---------------------%
c | dmdul is too small. |
c | Exit to avoid |
c | overflow. |
c %---------------------%
c
info = -15
go to 9000
end if
c
else
c
c %---------------------------%
c | Get the second eigenvalue |
c | of the conjugate pair by |
c | taking the conjugate of |
c | previous one. |
c %---------------------------%
c
dr(j) = dr(j-1)
di(j) = -di(j-1)
first = .true.
c
end if
c
280 continue
c
end if
c
end if
c
end if
c
end if
c
go to 9000
c
end if
c
c %----------------------------------------%
c | L O O P B A C K to call DNAUPD again. |
c %----------------------------------------%
c
go to 90
c
9000 continue
c
end
