nitgm2 Interface

Dimension of the problem

Array of length n containing the current $x$ value

Array of length n containing current approximate solution

Vector of of length n containing trial step

Relative residual reduction factor

User-supplied subroutine for evaluating the function the zero of which is sought.

User-supplied subroutine for optionally evaluating $J\vec{v}$ or $P^{-1}\vec{v}$, where $J$ is the Jacobian of $f$ and $P$ is a right preconditioning operator. If neither analytic $J\vec{v}$ evaluations nor right preconditioning is used, this can be a dummy subroutine; if right preconditioning is used but not analytic $J\vec{v}$ evaluations, this need only evaluate $P^{-1}\vec{v}$.

Parameter/work array passed to the f and jacv routines

Parameter/work array passed to the f and jacv routines

Flag for determining method of $J\vec{v}$ evaluation. 0 (default) indicates finite-difference evaluation, while 1 indicates analytic evaluation.

Flag for right preconditioning. 0 indicates no preconditioning, while 1 inidcates right preconditioning.

Maximum allowable number of GMRES iterations

Residual update flag. On GMRES restarts, the residual can be updated using a linear combination (iresup == 0) or by direct evaluation (iresup == 1). The first is cheap (one n-vector saxpy) but may lose accuracy with extreme residual reduction; the second retains accuracy better but costs one $J\vec{v}$ product.

Order of the finite-difference formula (sometimes) used on GMRES restarts when $J\vec{v}$ products are evaluated using finite- differences. When ijacv = 0 on input to nitsol, ifdord is set to 1, 2, or 4 in nitsol; otherwise, it is irrelevant. When ijacv = 0 on input to this subroutine, the precise meaning is as follows:

With GMRES, ifdord matters only if iresup = 1, in which case
it determines the order of the finite-difference formula used in evaluating the initial residual at each GMRES restart
(default 2). If iresup = 1 and ijacv = 0 on input to this
subroutine, then ijacv is temporarily reset to -1 at each
restart below to force a finite-difference evaluation of order ifdord. NOTE: This only affects initial residuals at restarts; first-order differences are always used within each GMRES
cycle. Using higher-order differences at restarts only should give the same accuracy as if higher-order differences were
used throughout; see K. Turner and H. F. Walker, "Efficient
high accuracy solutions with GMRES(m)," SIAM J. Sci. Stat.
Comput., 13 (1992), pp. 815--825.

Number of function evaluations

Number of $J\vec{v}$ evaluations

Number of $P^{-1}\vec{v}$ evaluations

Number of linear iterations

Maximum Krylov subspace dimension; default 10.

kdmax + 1

Matrix for storage of Krylov basis in GMRES; on return, the residual vector is contained in the first column.

Matrix for storage of triangular matrix in GMRES.

Vector for storage of estimate of singular vector of rr with largest singular value.

Vector for storage of estimate of singular vector of rr with smallest singular value.

Vector containing right-hand side of triangular system and least-squares residual norm in GMRES.

Work array

GMRES residual norm on return

Inner-product routine, either user-supplied or BLAS ddot.

Norm routine, either user supplied or BLAS dnrm2.

Termination flag. Values have the following meanings:

0

normal termination: acceptable step found

1

$J\vec{v}$ failure in nitjv

2

$P^{-1}\vec{v}$ failure in nitjv

3

Acceptable step not found in iksmax GMRES iterations

4

Insignificant residual norm reduction of a cycle of kdmax steps (stagnation) before an acceptable step has been found.

5

Dangerous ill-conditioning detected before an acceptable step has been found.