Nag library routine document d02cjf

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Nag library routine document d02cjf

Program Text d02bhfe. Program Results d02bhfe. D02BHF integrates a system of first-order ordinary differential equations over an interval with suitable initial conditions, using a Runge—Kutta—Merson method, until a user-specified function of the solution is zero.

The position of this sign change is then determined accurately by interpolating for the solution and its derivative. For a description of Runge—Kutta methods and their practical implementation see Hall and Watt Hall G and Watt J M ed.

On entry : must be set to the initial value of the independent variable x.

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On entry : the final value of the independent variable x. On entry : nthe number of differential equations. On entry : the initial values of the solution y 1y 2…y n. D02BHF has been designed so that, for most problems, a reduction in TOL leads to an approximately proportional reduction in the error in the solution obtained in the integration. However, the actual relation between TOL and the accuracy cannot be guaranteed.

You are strongly recommended to call D02BHF with more than one value for TOL and to compare the results obtained to estimate their accuracy. On exit : normally unchanged. On entry : determines the type of error control.

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At each step in the numerical solution an estimate of the local error, estis made. If the appropriate condition is not satisfied, the step size is reduced and the solution recomputed on the current step. It should be borne in mind that the computed solution will be used in evaluating g xy. The specification of FCN is:. Parameters denoted as Input must not be changed by this procedure.

The specification of G is:. On entry : xthe value of the independent variable. If you are unfamiliar with this parameter you should refer to Section 3. Otherwise, if you are not familiar with this parameter, the recommended value is 0. A serious error has occurred in an internal call to the specified routine.

Check all subroutine calls and array dimensions. Seek expert help. A serious error has occurred in an internal call to an integration routine. A serious error has occurred in an internal call to an interpolation routine. Check all sub program calls and array dimensions. An unexpected error has been triggered by this routine. Please contact NAG. See Section 3.

NAG Library Chapter Introduction

The accuracy may also be restricted by the properties of g xy. You should try to code G without introducing any unnecessary cancellation errors. The time taken by D02BHF depends on the complexity and mathematical properties of the system of differential equations defined by FCNthe complexity of Gon the range, the position of the solution and the tolerance. If the accuracy requested is really needed and cannot be obtained with this routine, the system may be very stiff see below or so badly scaled that it cannot be solved to the required accuracy.

This can happen for well-behaved systems and very small values of TOL. You should, however, consider whether there is a more fundamental difficulty. To determine whether a problem is stiff, D02PEF may be used.Routines are provided to work with Q in this representation see Section 8. Constraint: M. Constraint: N. On entry: the m by n matrix A. On exit: ifm n, the elements above the diagonal are overwritten by details of the orthogonal matrix Q and the lower triangle is overwritten by the corresponding elements of the m by m lower triangular matrix L.

On exit: further details of the orthogonal matrix Q. An explanatory message is output, and execution of the program is terminated. Program Text Program f8ahfe! Example Program Text! NAG Copyright !. Use Statements. Implicit None Statement Local Scalars. Local Arrays. Executable Statements. Compute the LQ factorization of A! The NAG name equivalent of dgelqf is f8ahf Call dgelqf m,n,a,lda,tau,work,lwork,info! Representation of a linear system. Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics.

A T A is positive. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues.

Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible. Section 1. All nonzero rows are above any rows of all zeros.Program Text d02bhfe. Program Results d02bhfe. C Header Interface include nagmk The position of this sign change is then determined accurately by interpolating for the solution and its derivative.

For a description of Runge—Kutta methods and their practical implementation see Hall and Watt Hall G and Watt J M ed. On entry : must be set to the initial value of the independent variable x. On entry : the final value of the independent variable x.

On entry : nthe number of differential equations. On entry : the initial values of the solution y 1y 2…y n.

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However, the actual relation between tol and the accuracy cannot be guaranteed. You are strongly recommended to call d02bhf with more than one value for tol and to compare the results obtained to estimate their accuracy. On exit : normally unchanged. On entry : determines the type of error control.

At each step in the numerical solution an estimate of the local error, estis made. If the appropriate condition is not satisfied, the step size is reduced and the solution recomputed on the current step. It should be borne in mind that the computed solution will be used in evaluating g xy. The specification of fcn is:. Arguments denoted as Input must not be changed by this procedure.

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Note: fcn should not return floating-point NaN Not a Number or infinity values, since these are not handled by d02bhf. If your code inadvertently does return any NaNs or infinities, d02bhf is likely to produce unexpected results. The specification of g is:.

On entry : xthe value of the independent variable.Program Results d02cjfe. D02CJF integrates a system of first-order ordinary differential equations over a range with suitable initial conditions, using a variable-order, variable-step Adams method until a user-specified function, if supplied, of the solution is zero, and returns the solution at points specified by you, if desired.

For a description of Adams methods and their practical implementation see Hall and Watt Hall G and Watt J M ed. On entry : the initial value of the independent variable x. On entry : the final value of the independent variable. On entry : nthe number of differential equations. The specification of FCN is:. On entry : xthe value of the independent variable.

Parameters denoted as Input must not be changed by this procedure. On entry : a positive tolerance for controlling the error in the integration. D02CJF has been designed so that, for most problems, a reduction in TOL leads to an approximately proportional reduction in the error in the solution. However, the actual relation between TOL and the accuracy achieved cannot be guaranteed.

You are strongly recommended to call D02CJF with more than one value for TOL and to compare the results obtained to estimate their accuracy.

nag library routine document d02cjf

On entry : the type of error control. At each step in the numerical solution an estimate of the local error, estis made. OUTPUT permits access to intermediate values of the computed solution for example to print or plot themat successive user-specified points. On entry : the output value of the independent variable x.

On entry : the computed solution at the point XSOL. The specification of G is:. If you do not require the root-finding option, the actual parameter G must be the dummy routine D02CJW.

If you are unfamiliar with this parameter you should refer to Section 3. Otherwise, if you are not familiar with this parameter, the recommended value is 0.

nag library routine document d02cjf

See Section 8 for a discussion of this error exit. A serious error has occurred in an internal call. Check all subroutine calls and array sizes. Seek expert help. The accuracy of the computation of the solution vector Y may be controlled by varying the local error tolerance TOL.Keyword Search:.

This chapter is concerned with the numerical solution of ordinary differential equations. There are two main types of problem: those in which all boundary conditions are specified at one point initial value problemsand those in which the boundary conditions are distributed between two or more points boundary value problems and eigenvalue problems. Routines are available for initial value problems, two-point boundary value problems and Sturm—Liouville eigenvalue problems.

A more general system may contain derivatives of higher order, but such systems can almost always be reduced to the first-order form by introducing new variables. It is sometimes advantageous to solve higher-order systems directly.

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There is also a boundary value routine that can treat directly a mixed order system of ordinary differential equations. There is a broader class of initial value problems known as differential algebraic systems which can be treated. Also a point, bat which the values of the dependent variables are required, must be specified. The routines in this chapter adjust the step length automatically to meet specified accuracy tolerances. Although the accuracy tests used are reliable over each step individually, in general an accuracy requirement cannot be guaranteed over a long range.

For many problems there may be no serious accumulation of error, but for unstable systems small perturbations of the solution will often lead to rapid divergence of the calculated values from the true values.

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A simple check for stability is to carry out trial calculations with different tolerances; if the results differ appreciably the system is probably unstable. Over a short range, the difficulty may possibly be overcome by taking sufficiently small tolerances, but over a long range it may be better to try to reformulate the problem. A special class of initial value problems are those for which the solutions contain rapidly decaying transient terms.

These problems require special methods for efficient numerical solution; the methods designed for non-stiff problems when applied to stiff problems tend to be very slow, because they need small step lengths to avoid numerical instability.

A full discussion is given in Hall and Watt and a discussion of the methods for stiff problems is given in Berzins et al. In general, a system of nonlinear differential equations with boundary conditions at two or more points cannot be guaranteed to have a solution. The solution, if it exists, has to be determined iteratively. A comprehensive treatment of the numerical solution of boundary value problems can be found in Ascher et al.

The methods for this chapter are discussed in Ascher et al. In the collocation method, the solution components are approximated by piecewise polynomials on a mesh. The coefficients of the polynomials form the unknowns to be computed. The approximation to the solution must satisfy the boundary conditions and the differential equations at collocation points in each mesh sub-interval. A modified Newton method is used to solve the nonlinear equations. The mesh is refined by trying to equidistribute the estimated error over the whole interval.

An initial estimate of the solution across the mesh is required. In the shooting method, the unknown boundary values at the initial point are estimated to form an initial value problem, and the equations are then integrated to the final point.

nag library routine document d02cjf

At the final point the computed solution and the known boundary conditions should be equal. The condition for equality gives a set of nonlinear equations for the estimated values, which can be solved by Newton's method or one of its variants. The iteration cannot be guaranteed to converge, but it is usually successful if the system has a solution, the system is not seriously unstable or very stiff for step-by-step solution, and good initial estimates can be found for the unknown boundary conditions.

It may be necessary to simplify the problem and carry out some preliminary calculations, in order to obtain suitable starting values. If a boundary value problem seems insoluble by the above method and a good estimate for the solution of the problem is known at all points of the range then a finite difference method may be used. Finite difference equations are set up on a mesh of points and estimated values for the solution at the grid points are chosen.

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nag library routine document d02cjf

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