“Newton’s method”的概念、定义、翻译、参考文献-科学参考

Newton’s method

Mathematics

The following method of finding successive approximations to a root of an equation f(x) = 0. Suppose that x₀ is a first approximation, known to be quite close to a root. If the root is in fact x₀ + h, where h is ‘small’, taking the first two terms of the Taylor series gives f(x₀ + h) ≈ f(x₀) + hf′(x₀). Since f(x₀ + h) = 0, it follows that h ≈ −f(x₀)/ f′(x₀). Thus x₁, given by

$x_{1} = x_{0} - \frac{f (x_{0})}{f' (x_{0})},$

is likely to be a better approximation to the root. To see the geometrical significance of the method, suppose that P₀ is the point (x₀, f(x₀)) on the curve y = f(x), as shown in the first figure. The value x₁ is given by the point at which the tangent to the curve at P₀ meets the x-axis. It may be possible to repeat the process to obtain successive approximations x₀, x₁, x₂,…, where

$x_{n + 1} = x_{n} - \frac{f (x_{n})}{f' (x_{n})} .$

Iterations converge to root

Iterations diverge

These may be successively better approximations to the root as required, but in the second figure is shown the graph of a function, with a value x₀ close to a root, for which x₁ and x₂ do not give successively better approximations. However, if f″(x) is a continuous function and f′(x) is non-zero, then Newton’s method quadratically converges to the root in some neighbourhood of the root.

For a system of k equations in k variables, f(x) = 0, the iteration becomes

$x_{n + 1} = x_{n} - J^{- 1} (x_{n}) f (x_{n})$

where J denotes the Jacobian matrix of f.

Computer

An iterative technique for solving one or more nonlinear equations. For the single equation
$f (x) = 0$
the iteration is
$\begin{array}{l} x_{n + 1} = x_{n} - f (x_{n}) / f' (x_{n}), \\ n = 0, 1, 2, \dots \end{array}$
where x₀ is an approximation to the solution. For the system
$\begin{array}{l} f (x) = 0, \\ f = {(f_{1}, f_{2}, \dots, f_{n})}^{T}, \\ x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}, \end{array}$
the iteration takes the mathematical form
$\begin{array}{l} x_{n} +_{1} = x_{n} - J {(x_{n})}^{- 1} f (x_{n}), \\ n = 0, 1, 2, \dots \end{array}$
where J(x) is the n′ n matrix whose i,jth element is
$\partial f_{i} (x) / \partial x_{j}$
In practice each iteration is carried out by solving a system of linear equations. Subject to appropriate conditions the iteration converges quadratically (ultimately an approximate squaring of the error occurs). The disadvantage of the method is that a constant recalculation of J may be too time-consuming and so the method is most often used in a modified form, e.g. with approximate derivatives. Since Newton’s method is derived by a linearization of f(x), it is capable of generalization to other kinds of nonlinear problems, e.g. boundary-value problems (see ordinary differential equations).

单词	Newton’s method
释义	Newton’s method Mathematics The following method of finding successive approximations to a root of an equation f(x) = 0. Suppose that x₀ is a first approximation, known to be quite close to a root. If the root is in fact x₀ + h, where h is ‘small’, taking the first two terms of the Taylor series gives f(x₀ + h) ≈ f(x₀) + hf′(x₀). Since f(x₀ + h) = 0, it follows that h ≈ −f(x₀)/ f′(x₀). Thus x₁, given by $x_{1} = x_{0} - \frac{f (x_{0})}{f' (x_{0})},$ is likely to be a better approximation to the root. To see the geometrical significance of the method, suppose that P₀ is the point (x₀, f(x₀)) on the curve y = f(x), as shown in the first figure. The value x₁ is given by the point at which the tangent to the curve at P₀ meets the x-axis. It may be possible to repeat the process to obtain successive approximations x₀, x₁, x₂,…, where $x_{n + 1} = x_{n} - \frac{f (x_{n})}{f' (x_{n})} .$ Iterations converge to root Iterations diverge These may be successively better approximations to the root as required, but in the second figure is shown the graph of a function, with a value x₀ close to a root, for which x₁ and x₂ do not give successively better approximations. However, if f″(x) is a continuous function and f′(x) is non-zero, then Newton’s method quadratically converges to the root in some neighbourhood of the root. For a system of k equations in k variables, f(x) = 0, the iteration becomes $x_{n + 1} = x_{n} - J^{- 1} (x_{n}) f (x_{n})$ where J denotes the Jacobian matrix of f. Computer An iterative technique for solving one or more nonlinear equations. For the single equation $f (x) = 0$ the iteration is $\begin{array}{l} x_{n + 1} = x_{n} - f (x_{n}) / f' (x_{n}), \\ n = 0, 1, 2, \dots \end{array}$ where x₀ is an approximation to the solution. For the system $\begin{array}{l} f (x) = 0, \\ f = {(f_{1}, f_{2}, \dots, f_{n})}^{T}, \\ x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}, \end{array}$ the iteration takes the mathematical form $\begin{array}{l} x_{n} +_{1} = x_{n} - J {(x_{n})}^{- 1} f (x_{n}), \\ n = 0, 1, 2, \dots \end{array}$ where J(x) is the n′ n matrix whose i,jth element is $\partial f_{i} (x) / \partial x_{j}$ In practice each iteration is carried out by solving a system of linear equations. Subject to appropriate conditions the iteration converges quadratically (ultimately an approximate squaring of the error occurs). The disadvantage of the method is that a constant recalculation of J may be too time-consuming and so the method is most often used in a modified form, e.g. with approximate derivatives. Since Newton’s method is derived by a linearization of f(x), it is capable of generalization to other kinds of nonlinear problems, e.g. boundary-value problems (see ordinary differential equations).
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