c05qsc is based on the MINPACK routine HYBRD1 (see Moré et al. (1980)). It chooses the correction at each step as a convex combination of the Newton and scaled gradient directions. The Jacobian is updated by the sparse rank-1 method of Schubert (see Schubert (1970)). At the starting point, the sparsity pattern is determined and the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. Then, the sparsity structure is used to recompute an approximation to the Jacobian by forward differences with the least number of function evaluations. The function you supply must be able to compute only the requested subset of the function values. The sparse Jacobian linear system is solved at each iteration with f11mec computing the Newton step. For more details see Powell (1970) and Broyden (1965).
4References
Broyden C G (1965) A class of methods for solving nonlinear simultaneous equations Mathematics of Computation19(92) 577–593
Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory
Powell M J D (1970) A hybrid method for nonlinear algebraic equations Numerical Methods for Nonlinear Algebraic Equations (ed P Rabinowitz) Gordon and Breach
Schubert L K (1970) Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian Mathematics of Computation24(109) 27–30
5Arguments
1: $\mathbf{fcn}$ – function, supplied by the userExternal Function
fcn must return the values of the functions ${f}_{i}$ at a point $x$.
On entry: indf specifies the indices $i$ for which values of ${f}_{i}\left(x\right)$ must be computed. The indices are specified in strictly ascending order.
On exit: ${\mathbf{fvec}}\left[i-1\right]$ must contain the function values ${f}_{i}\left(x\right)$, for all indices $i$ in indf.
6: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fcn.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling c05qsc you may allocate memory and initialize these pointers with various quantities for use by fcn when called from c05qsc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
7: $\mathbf{iflag}$ – Integer *Input/Output
On entry: ${\mathbf{iflag}}>0$.
On exit: in general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer.
Note:fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05qsc. If your code inadvertently does return any NaNs or infinities, c05qsc is likely to produce unexpected results.
On exit: the function values at the final point returned in x. ${\mathbf{fvec}}\left[i-1\right]$ contains the function values ${f}_{i}$.
5: $\mathbf{xtol}$ – doubleInput
On entry: the accuracy in x to which the solution is required.
Suggested value:
$\sqrt{\epsilon}$, where $\epsilon $ is the machine precision returned by X02AJC.
Constraint:
${\mathbf{xtol}}\ge 0.0$.
6: $\mathbf{init}$ – Nag_BooleanInput
On entry: init must be set to Nag_TRUE to indicate that this is the first time c05qsc is called for this specific problem. c05qsc then computes the dense Jacobian and detects and stores its sparsity pattern (in rcomm and icomm) before proceeding with the iterations. This is noticeably time consuming when n is large. If not enough storage has been provided for rcomm or icomm, c05qsc will fail. On exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL, ${\mathbf{icomm}}\left[0\right]$ contains $\mathit{nnz}$, the number of nonzero entries found in the Jacobian. On subsequent calls, init can be set to Nag_FALSE if the problem has a Jacobian of the same sparsity pattern. In that case, the computation time required for the detection of the sparsity pattern will be smaller.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_NO_IMPROVEMENT, NE_TOO_MANY_FEVALS or NE_TOO_SMALL on exit, ${\mathbf{icomm}}\left[0\right]$ contains $\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian.
icomm MUST NOT be altered between successive calls to c05qsc.
Constraint:
${\mathbf{licomm}}\ge 8\times {\mathbf{n}}+19+\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian, as computed by c05qsc.
11: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
12: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{licomm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{licomm}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{lrcomm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lrcomm}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_IMPROVEMENT
The iteration is not making good progress. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05qsc from a different starting point may avoid the region of difficulty. The condition number of the Jacobian is $\u27e8\mathit{\text{value}}\u27e9$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least $200\times ({\mathbf{n}}+1)$ calls to fcn. Consider setting ${\mathbf{init}}=\mathrm{Nag\_FALSE}$ and restarting the calculation from the point held in x.
NE_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
If this condition is satisfied with ${\mathbf{xtol}}={10}^{-k}$, then the larger components of $x$ have $k$ significant decimal digits. There is a danger that the smaller components of $x$ may have large relative errors, but the fast rate of convergence of c05qsc usually obviates this possibility.
If xtol is less than machine precision and the above test is satisfied with the machine precision in place of xtol, then the function exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_TOO_SMALL.
Note: this convergence test is based purely on relative error, and may not indicate convergence if the solution is very close to the origin.
The convergence test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then c05qsc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qsc with a lower value for xtol.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c05qsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
Local workspace arrays of fixed lengths are allocated internally by c05qsc. The total size of these arrays amounts to $8\times n+2\times q$ double elements and $10\times n+2\times q+5$ integer elements where the integer $q$ is bounded by $8\times \mathit{nnz}$ and ${n}^{2}$ and depends on the sparsity pattern of the Jacobian.
The time required by c05qsc to solve a given problem depends on $n$, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by c05qsc to process each evaluation of the functions depends on the number of nonzero entries in the Jacobian. The timing of c05qsc is strongly influenced by the time spent evaluating the functions.
When init is Nag_TRUE, the dense Jacobian is first evaluated and that will take time proportional to ${n}^{2}$.
Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.
10Example
This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations: