<?xml version="1.0" encoding="UTF-8"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="dnaupd" xml:lang="ja">
    <refnamediv>
        <refname>dnaupd</refname>
        <refpurpose>
            暗黙のうちに再開されるArnoldi反復へのインターフェイスで,
            実線形演算子の小数の固有値/ベクトルの組を近似的に計算します.
            <emphasis role="bold">
                この関数は廃止されました. <link linkend="eigs">eigs</link>を使用してください
            </emphasis>
        </refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[IDO, RESID, V, IPARAM, IPNTR, WORKD, WORKL, INFO] = dnaupd(ID0, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, IPARAM, IPNTR, WORKD, WORKL, INFO)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>ID0</term>
                <listitem>
                    <para>
                        Integer. (INPUT/OUTPUT)
                    </para>
                    <para>
                        Reverse communication flag. IDO must
                        be zero on the first call to dnaupd. IDO will be set internally to
                        indicate the type of operation to be performed. Control is then
                        given back to the calling routine which has the responsibility to
                        carry out the requested operation and call dnaupd with the result.
                        The operand is given in WORKD(IPNTR(1)), the result must be put in
                        WORKD(IPNTR(2)).
                    </para>
                    <itemizedlist>
                        <listitem>
                            <para>
                                IDO = 0: first call to the reverse communication
                                interface.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IDO = -1: compute Y = OP * X where IPNTR(1) is the pointer
                                into WORKD for X, IPNTR(2) is the pointer into WORKD for Y. This
                                is for the initialization phase to force the starting vector
                                into the range of OP.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IDO = 1: compute Y = OP * X where IPNTR(1) is the pointer
                                into WORKD for X, IPNTR(2) is the pointer into WORKD for Y. In
                                mode 3 and 4, the vector B * X is already available in
                                WORKD(ipntr(3)). It does not need to be recomputed in forming OP
                                * X.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IDO = 2: compute Y = B * X where IPNTR(1) is the pointer
                                into WORKD for X, IPNTR(2) is the pointer into WORKD for
                                Y.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IDO = 3: compute the IPARAM(8) real and imaginary parts of
                                the shifts where INPTR(14) is the pointer into WORKL for placing
                                the shifts. See Remark 5 below.
                            </para>
                        </listitem>
                        <listitem>
                            <para>IDO = 99: done.</para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>BMAT</term>
                <listitem>
                    <para>
                        Character. (INPUT)
                    </para>
                    <para>
                        specifies the type of the matrix B that defines the
                        semi-inner product for the operator OP.
                    </para>
                    <itemizedlist>
                        <listitem>
                            <para>'I' - standard eigenvalue problem A * x = lambda * x</para>
                        </listitem>
                        <listitem>
                            <para>
                                'G' - generalized eigenvalue problem A * x =
                                lambda * B * x
                            </para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>N</term>
                <listitem>
                    <para>Integer. (INPUT)</para>
                    <para>dimension of the eigenproblem.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>WHICH</term>
                <listitem>
                    <para>
                        string of length 2. (INPUT)
                    </para>
                    <para>
                        Specifies which of the Ritz values of OP to
                        compute.
                    </para>
                    <itemizedlist>
                        <listitem>
                            <para>
                                'LM' - want the NEV eigenvalues of largest
                                magnitude.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                'SM' - want the NEV eigenvalues of smallest
                                magnitude.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                'LR' - want the NEV eigenvalues of largest real
                                part.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                'SR' - want the NEV eigenvalues of smallest real
                                part.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                'LI' - want the NEV eigenvalues of largest imaginary
                                part.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                'SI' - want the NEV eigenvalues of smallest imaginary
                                part.
                            </para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>NEV</term>
                <listitem>
                    <para>
                        Integer. (INPUT)
                    </para>
                    <para>
                        number of eigenvalues of OP to be computed. 0 &lt;
                        NEV &lt; N-1.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>TOL</term>
                <listitem>
                    <para>
                        scalar. (INPUT)
                    </para>
                    <para>
                        Stopping criterion: the relative accuracy of the Ritz
                        value is considered acceptable if BOUNDS(I) &lt;= TOL*ABS(RITZ(I)).
                        If TOL &lt;= 0. is passed the machine precision is set.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>RESID</term>
                <listitem>
                    <para>array of length N. (INPUT/OUTPUT)</para>
                    <para>
                        On INPUT: If INFO = 0, a random initial residual vector is
                        used, else RESID contains the initial residual vector, possibly from
                        a previous run.
                    </para>
                    <para>On OUTPUT: RESID contains the final residual vector.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>NCV</term>
                <listitem>
                    <para>
                        Integer. (INPUT)
                    </para>
                    <para>
                        number of columns of the matrix V. NCV must satisfy
                        the two inequalities 2 &lt;= NCV - NEV and NCV &lt;= N.
                    </para>
                    <para>
                        This will indicate how many Arnoldi vectors are generated at
                        each iteration.
                    </para>
                    <para>
                        After the startup phase in which NEV Arnoldi vectors are
                        generated, the algorithm generates approximately NCV - NEV Arnoldi
                        vectors at each subsequent update iteration. Most of the cost in
                        generating each Arnoldi vector is in the matrix-vector operation
                        OP * x.
                    </para>
                    <para>
                        NOTE: 2 &lt;= NCV - NEV in order that complex conjugate pairs of
                        Ritz values are kept together. (See remark 4 below)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>V</term>
                <listitem>
                    <para>
                        N by NCV array. (OUTPUT)
                    </para>
                    <para>
                        Contains the final set of Arnoldi basis
                        vectors.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>IPARAM</term>
                <listitem>
                    <para>array of length 11. (INPUT/OUTPUT)</para>
                    <itemizedlist>
                        <listitem>
                            <para>
                                IPARAM(1) = ISHIFT: method for selecting the implicit
                                shifts. The shifts selected at each iteration are used to
                                restart the Arnoldi iteration in an implicit fashion.
                            </para>
                            <itemizedlist>
                                <listitem>
                                    <para>
                                        ISHIFT = 0: the shifts are provided by the user via
                                        reverse communication. The real and imaginary parts of the
                                        NCV eigenvalues of the Hessenberg matrix H are returned in
                                        the part of the WORKL array corresponding to RITZR and
                                        RITZI. See remark 5 below.
                                    </para>
                                </listitem>
                                <listitem>
                                    <para>
                                        ISHIFT = 1: exact shifts with respect to the current
                                        Hessenberg matrix H. This is equivalent to restarting the
                                        iteration with a starting vector that is a linear
                                        combination of approximate Schur vectors associated with the
                                        "wanted" Ritz values.
                                    </para>
                                </listitem>
                            </itemizedlist>
                        </listitem>
                        <listitem>
                            <para>IPARAM(2) = LEVEC. No longer referenced.</para>
                        </listitem>
                        <listitem>
                            <para>IPARAM(3) = MXITER </para>
                            <para>
                                On INPUT: maximum number of Arnoldi update iterations
                                allowed.
                            </para>
                            <para>
                                On OUTPUT: actual number of Arnoldi update iterations
                                taken.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPARAM(4) = NB: blocksize to be used in the recurrence.
                                The code currently works only for NB = 1.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPARAM(5) = NCONV: number of "converged" Ritz values. This
                                represents the number of Ritz values that satisfy the
                                convergence criterion.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPARAM(6) = IUPD No longer referenced. Implicit restarting
                                is ALWAYS used.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPARAM(7) = MODE On INPUT determines what type of
                                eigenproblem is being solved. Must be 1,2,3,4; See under
                                Description of dnaupd for the five modes available.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPARAM(8) = NP When ido = 3 and the user provides shifts
                                through reverse communication (IPARAM(1)=0), dnaupd returns NP,
                                the number of shifts the user is to provide.
                            </para>
                            <para>0 &lt; NP &lt;= NCV-NEV. See Remark 5 below.</para>
                        </listitem>
                        <listitem>
                            <para>IPARAM(9) = NUMOP, </para>
                            <para>IPARAM(10) = NUMOPB, </para>
                            <para>IPARAM(11) = NUMREO, </para>
                            <para>
                                On OUTPUT: NUMOP = total number of OP*x operations, NUMOPB
                                = total number of B*x operations if BMAT='G', NUMREO = total
                                number of steps of re-orthogonalization.
                            </para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>IPNTR</term>
                <listitem>
                    <para>
                        array of length 14. (OUTPUT)
                    </para>
                    <para>
                        Pointer to mark the starting locations in
                        the WORKD and WORKL arrays for matrices/vectors used by the Arnoldi
                        iteration.
                    </para>
                    <itemizedlist>
                        <listitem>
                            <para>
                                IPNTR(1): pointer to the current operand vector X in
                                WORKD.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(2): pointer to the current result vector Y in
                                WORKD.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(3): pointer to the vector B * X in WORKD when used
                                in the shift-and-invert mode.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(4): pointer to the next available location in WORKL
                                that is untouched by the program.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(5): pointer to the NCV by NCV upper Hessenberg
                                matrix H in WORKL.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(6): pointer to the real part of the ritz value array
                                RITZR in WORKL.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(7): pointer to the imaginary part of the ritz value
                                array RITZI in WORKL.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(8): pointer to the Ritz estimates in array WORKL
                                associated with the Ritz values located in RITZR and RITZI in
                                WORKL.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5
                                below.
                            </para>
                        </listitem>
                    </itemizedlist>
                    <para>
                        Note: IPNTR(9:13) is only referenced by dneupd . See Remark
                        2.
                    </para>
                    <itemizedlist>
                        <listitem>
                            <para>
                                IPNTR(9): pointer to the real part of the NCV RITZ values
                                of the original system.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(10): pointer to the imaginary part of the NCV RITZ
                                values of the original system.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(11): pointer to the NCV corresponding error
                                bounds.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(12):pointer to the NCV by NCV upper quasi-triangular
                                Schur matrix for H.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                IPNTR(13): pointer to the NCV by NCV matrix of
                                eigenvectors of the upper Hessenberg matrix H. Only referenced
                                by dneupd if RVEC = 1 See Remark 2 below.
                            </para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>WORKD</term>
                <listitem>
                    <para>
                        Double precision work array of length 3 * N. (REVERSE
                        COMMUNICATION)
                    </para>
                    <para>
                        Distributed array to be used in the basic Arnoldi iteration
                        for reverse communication. The user should not use WORKD as
                        temporary workspace during the iteration. Upon termination
                        WORKD(1:N) contains B*RESID(1:N). If an invariant subspace
                        associated with the converged Ritz values is desired, see remark 2
                        below, subroutine dneupd uses this output. See Data Distribution
                        Note below.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>WORKL</term>
                <listitem>
                    <para>
                        work array of length at least 3 * NCV ** 2 + 6 * NCV.
                        (OUTPUT/WORKSPACE)
                    </para>
                    <para>
                        Private (replicated) array on each PE or array
                        allocated on the front end. See Data Distribution Note below.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>INFO</term>
                <listitem>
                    <para>Integer. (INPUT/OUTPUT)</para>
                    <para>
                        If INFO == 0, a randomly initial residual vector is used, else
                        RESID contains the initial residual vector, possibly from a previous
                        run.
                    </para>
                    <para>Error flag on output.</para>
                    <itemizedlist>
                        <listitem>
                            <para>0: Normal exit.</para>
                        </listitem>
                        <listitem>
                            <para>
                                1: Maximum number of iterations taken. All possible
                                eigenvalues of OP has been found. IPARAM(5) returns the number
                                of wanted converged Ritz values.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                2: No longer an informational error. Deprecated starting
                                with release 2 of ARPACK.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                3: No shifts could be applied during a cycle of the
                                Implicitly restarted Arnoldi iteration. One possibility is to
                                increase the size of NCV relative to NEV. See remark 4
                                below.
                            </para>
                        </listitem>
                        <listitem>
                            <para>-1: N must be positive.</para>
                        </listitem>
                        <listitem>
                            <para>-2: NEV must be positive.</para>
                        </listitem>
                        <listitem>
                            <para>-3: NCV-NEV &gt;= 2 and less than or equal to N.</para>
                        </listitem>
                        <listitem>
                            <para>
                                -4: The maximum number of Arnoldi update iterations
                                allowed must be greater than zero.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI',
                                'SI'.
                            </para>
                        </listitem>
                        <listitem>
                            <para>-6: BMAT must be one of 'I' or 'G'.</para>
                        </listitem>
                        <listitem>
                            <para>
                                -7: Length of private work array WORKL is not
                                sufficient.
                            </para>
                        </listitem>
                        <listitem>
                            <para>
                                -8: Error return from LAPACK eigenvalue
                                calculation.
                            </para>
                        </listitem>
                        <listitem>
                            <para>-9: Starting vector is zero.</para>
                        </listitem>
                        <listitem>
                            <para>-10: IPARAM(7) must be 1, 2, 3, 4.</para>
                        </listitem>
                        <listitem>
                            <para>-11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.</para>
                        </listitem>
                        <listitem>
                            <para>-12: IPARAM(1) must be equal to 0 or 1.</para>
                        </listitem>
                        <listitem>
                            <para>
                                -9999: Could not build an Arnoldi factorization. IPARAM(5)
                                returns the size of the current Arnoldi factorization. The user
                                is advised to check that enough workspace and array storage has
                                been allocated.
                            </para>
                        </listitem>
                    </itemizedlist>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Reverse communication interface for the Implicitly Restarted Arnoldi
            iteration. This subroutine computes approximations to a few eigenpairs of
            a linear operator "OP" with respect to a semi-inner product defined by a
            symmetric positive semi-definite real matrix B. B may be the identity
            matrix. NOTE: If the linear operator "OP" is real and symmetric with
            respect to the real positive semi-definite symmetric matrix B, i.e. B*OP =
            (OP`)*B, then subroutine dsaupd should be used instead.
        </para>
        <para>
            The computed approximate eigenvalues are called Ritz values and the
            corresponding approximate eigenvectors are called Ritz vectors.
        </para>
        <para>
            dnaupd is usually called iteratively to solve one of the following
            problems:
        </para>
        <itemizedlist>
            <listitem>
                <para>
                    Mode 1: A*x = lambda*x. <literal>
                        OP = A , B =
                        I
                    </literal>
                    .
                </para>
            </listitem>
            <listitem>
                <para>
                    Mode 2: A*x = lambda*M*x, M symmetric positive definite
                    <literal>OP = inv[M]*A, B = M</literal>. (If M can be factored see
                    remark 3 below)
                </para>
            </listitem>
            <listitem>
                <para>
                    Mode 3: A*x = lambda*M*x, M symmetric positive semi-definite.
                    <literal>OP = Real_Part{ inv[A - sigma*M]*M }, B = M</literal>.
                    shift-and-invert mode (in real arithmetic)
                </para>
                <para>
                    If <literal>OP*x = amu*x</literal>, then
                </para>
                <para>
                    <literal>
                        amu = 1/2 * [ 1/(lambda-sigma) +
                        1/(lambda-conjg(sigma))]
                    </literal>
                    .
                </para>
                <para>
                    Note: If sigma is real, i.e. imaginary part of sigma is zero;
                    <literal>
                        Real_Part{ inv[A - sigma*M]*M } == inv[A -
                        sigma*M]*M
                    </literal>
                    <literal>
                        amu ==
                        1/(lambda-sigma)
                    </literal>
                    .
                </para>
            </listitem>
            <listitem>
                <para>
                    Mode 4: A*x = lambda*M*x, M symmetric semi-definite <literal>
                        OP
                        = Imaginary_Part{ inv[A - sigma*M]*M } , B = M
                    </literal>
                    .
                    shift-and-invert mode (in real arithmetic)
                </para>
                <para>
                    If <literal>OP*x = amu*x</literal>, then <literal>
                        amu = 1/2i * [
                        1/(lambda-sigma) - 1/(lambda-conjg(sigma)) ]
                    </literal>
                    .
                </para>
            </listitem>
        </itemizedlist>
        <para>
            Both mode 3 and 4 give the same enhancement to eigenvalues close to
            the (complex) shift sigma. However, as lambda goes to infinity, the
            operator OP in mode 4 dampens the eigenvalues more strongly than does OP
            defined in mode 3.
        </para>
        <para>
            NOTE: The action of w &lt;- inv[A - sigma * M] * v or w &lt;- inv[M] * v
            should be accomplished either by a direct method using a sparse matrix
            factorization and solving <literal>[A - sigma * M] * w = v</literal> or
            <literal>M * w = v</literal>, or through an iterative method for solving
            these systems. If an iterative method is used, the convergence test must
            be more stringent than the accuracy requirements for the eigenvalue
            approximations.
        </para>
    </refsection>
    <refsection>
        <title>Example</title>
        <programlisting role="example">
            <![CDATA[
// The following sets dimensions for this problem.
nx    = 10;
nev   = 3;
ncv   = 6;
bmat  = 'I';
which = 'LM';
// Local Arrays
iparam  = zeros(11, 1);
ipntr   = zeros(14, 1);
_select = zeros(ncv, 1);
dr       = zeros(nev + 1, 1);
di      = zeros(nev + 1, 1);
z       = zeros(nx, nev + 1);
resid   = zeros(nx, 1);
v       = zeros(nx, ncv);
workd   = zeros(3 * nx, 1);
workev  = zeros(3 * ncv, 1);
workl   = zeros(3 * ncv * ncv + 6 * ncv, 1);
// Build the test matrix
A            = diag(10 * ones(nx, 1));
A(1:$-1,2:$) = A(1:$-1,2:$) + diag(6 * ones(nx-1,1));
A(2:$,1:$-1) = A(2:$,1:$-1) + diag(-6 * ones(nx-1,1));
tol    = 0;
ido    = 0;
ishfts = 1;
maxitr = 300;
mode1  = 1;
iparam(1) = ishfts;
iparam(3) = maxitr;
iparam(7) = mode1;
sigmar = 0; // the real part of the shift
sigmai = 0; // the imaginary part of the shift
info_dnaupd = 0;
// M A I N   L O O P (Reverse communication)
while(ido <> 99)
  // Repeatedly call the routine DNAUPD and take actions indicated by parameter IDO until
  // either convergence is indicated or maxitr has been exceeded.
  [ido, resid, v, iparam, ipntr, workd, workl, info_dnaupd] = dnaupd(ido, bmat, nx, which, nev, tol, resid, ncv, v, iparam, ipntr, workd, workl, info_dnaupd);
  if(info_dnaupd < 0)
    printf('\nError with dnaupd, info = %d\n',info_dnaupd);
    printf('Check the documentation of dnaupd\n\n');
  end
  if(ido == -1 | ido == 1)
    // Perform matrix vector multiplication
    workd(ipntr(2):ipntr(2) + nx -1) = A * workd(ipntr(1):ipntr(1) + nx - 1);
  end
end
// Post-Process using DNEUPD.
rvec    = 1;
howmany = 'A';
info_dneupd = 0;
[dr, di, z, resid, v, iparam, ipntr, workd, workl, info_dneupd] = dneupd(rvec, howmany, _select, dr, di, z, sigmar, sigmai, workev, ...
                                                                       bmat, nx, which, nev, tol, resid, ncv, v, ...
                                                                       iparam, ipntr, workd, workl, info_dneupd);
if(info_dneupd < 0)
  printf('\nError with dneupd, info = %d\n', info_dneupd);
  printf('Check the documentation of dneupd.\n\n');
end
printf('\nDNSIMP\n');
printf('======\n');
printf('\n');
printf('Size of the matrix is %d\n', nx);
printf('The number of Ritz values requested is %d\n', nev);
printf('The number of Arnoldi vectors generated (NCV) is %d\n', ncv);
printf('What portion of the spectrum: %s\n', which);
printf('The number of Implicit Arnoldi update iterations taken is %d\n', iparam(3));
printf('The number of OP*x is %d\n', iparam(9));
printf('The convergence criterion is %d\n', tol);
]]>
        </programlisting>
    </refsection>
    <refsection>
        <title>Remarks</title>
        <para>
            1. The computed Ritz values are approximate eigenvalues of OP. The
            selection of WHICH should be made with this in mind when Mode = 3 and 4.
            After convergence, approximate eigenvalues of the original problem may be
            obtained with the ARPACK subroutine dneupd.
        </para>
        <para>
            2. If a basis for the invariant subspace corresponding to the
            converged Ritz values is needed, the user must call dneupd immediately
            following completion of dnaupd. This is new starting with release 2 of
            ARPACK.
        </para>
        <para>
            3. If M can be factored into a Cholesky factorization M = LL` then
            Mode = 2 should not be selected. Instead one should use Mode = 1 with OP =
            inv(L) * A * inv(L`). Appropriate triangular linear systems should be solved
            with L and L` rather than computing inverses. After convergence, an
            approximate eigenvector z of the original problem is recovered by solving
            L`z = x where x is a Ritz vector of OP.
        </para>
        <para>
            4. At present there is no a-priori analysis to guide the selection
            of NCV relative to NEV. The only formal requirement is that NCV &gt; NEV +
            2. However, it is recommended that NCV &gt;= 2 * NEV + 1. If many problems of
            the same type are to be solved, one should experiment with increasing NCV
            while keeping NEV fixed for a given test problem. This will usually
            decrease the required number of OP*x operations but it also increases the
            work and storage required to maintain the orthogonal basis vectors. The
            optimal "cross-over" with respect to CPU time is problem dependent and
            must be determined empirically. See Chapter 8 of Reference 2 for further
            information.
        </para>
        <para>
            5. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the NP
            = IPARAM(8) real and imaginary parts of the shifts in locations
        </para>
        <programlisting>
            real part                  imaginary part
            -----------------------    --------------
            1   WORKL(IPNTR(14))           WORKL(IPNTR(14) + NP)
            2   WORKL(IPNTR(14) + 1)         WORKL(IPNTR(14) + NP + 1)
            .                          .
            .                          .
            .                          .
            NP  WORKL(IPNTR(14) + NP - 1)      WORKL(IPNTR(14) + 2 * NP - 1).
        </programlisting>
        <para>
            Only complex conjugate pairs of shifts may be applied and the pairs
            must be placed in consecutive locations. The real part of the eigenvalues
            of the current upper Hessenberg matrix are located in WORKL(IPNTR(6))
            through WORKL(IPNTR(6) + NCV - 1) and the imaginary part in WORKL(IPNTR(7))
            through WORKL(IPNTR(7) + NCV - 1). They are ordered according to the order
            defined by WHICH. The complex conjugate pairs are kept together and the
            associated Ritz estimates are located in WORKL(IPNTR(8)),
            WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8) + NCV - 1).
        </para>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="dsaupd">dsaupd</link>
            </member>
            <member>
                <link linkend="dneupd">dneupd</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Bibliography</title>
        <para>
            1. D.C. Sorensen, "Implicit Application of Polynomial Filters in a
            k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992), pp
            357-385.
        </para>
        <para>
            2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly
            Restarted Arnoldi Iteration", Rice University Technical Report TR95-13,
            Department of Computational and Applied Mathematics.
        </para>
        <para>
            3. B.N. Parlett, "The Symmetric Eigenvalue Problem". Prentice-Hall,
            1980.
        </para>
        <para>
            4. B.N. Parlett, B. Nour-Omid, "Towards a Black Box Lanczos
            Program", Computer Physics Communications, 53 (1989), pp 169-179.
        </para>
        <para>
            5. B. Nour-Omid, B.N. Parlett, T. Ericson, P.S. Jensen, "How to
            Implement the Spectral Transformation", Math. Comp., 48 (1987), pp
            663-673.
        </para>
        <para>
            6. R.G. Grimes, J.G. Lewis and H.D. Simon, "A Shifted Block Lanczos
            Algorithm for Solving Sparse Symmetric Generalized Eigenproblems", SIAM J.
            Matr. Anal. Apps., January (1993).
        </para>
        <para>
            7. L. Reichel, W.B. Gragg, "Algorithm 686: FORTRAN Subroutines for
            Updating the QR decomposition", ACM TOMS, December 1990, Volume 16 Number
            4, pp 369-377.
        </para>
        <para>
            8. R.B. Lehoucq, D.C. Sorensen, "Implementation of Some Spectral
            Transformations in a k-Step Arnoldi Method". In Preparation.
        </para>
    </refsection>
    <refsection>
        <title>Used Functions</title>
        <para>Based on ARPACK routine dnaupd</para>
    </refsection>
    <refsection>
        <title>履歴</title>
        <revhistory>
            <revision>
                <revnumber>5.4.0</revnumber>
                <revremark>
                    関数は廃止され,<link linkend="eigs">eigs</link>に代替されました.
                </revremark>
            </revision>
        </revhistory>
    </refsection>
</refentry>
