Use an approximation as if it were exact

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Use an approximation as if it were exact

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[QUICK DESCRIPTION]
Often we want to apply a method to a general class of functions, but we only know how to do this for a certain class of functions (e.g., polynomials).  Then use an approximation to our general function from our certain class, and act as if our approximation were exact.

A more sophisticated version, is to use this to ''update'' a guess by means of this approach, obtaining successively (we hope) approximations to the unknown we wish to compute.

[PREREQUISITES]
Linear algebra, calculus.

[EXAMPLE]
Numerical integration is mostly done this way.

We take a polynomial interpolant $p$ of a given function $f:[a,b]\to\R$ at certain chosen points $x_1<x_2<\cdots<x_N$ and then use $\int_a^b p(x)\,dx$ as the approximation to $\int_a^b f(x)\,dx$.

The error in the result is usually best estimated by the error in the interpolant, or by noting that the method is exact for all polynomials of up to a certain degree (at least $N-1$), and then using the error in the interpolant to estimate the error in the integral.  (See also [[Interpolation and approximation]].)

[EXAMPLE]
Solving linear or nonlinear equations iteratively.

For a linear system of equations, $Ax=b$ where we wish to find $x$, if we have an approximate matrix $B\approx A$ for which we can readily solve systems $Bz=d$, we can compute $x$ iteratively via:
[maths]\begin{align} B\delta_i&=b-Ax_i\\
                     x_{i+1}  &=x_i + \delta_i.
\end{align}[/maths]
This is known as ''iterative refinement''.

For nonlinear systems we use the Taylor series approximation to 1st order: $f(x+\delta)=f(x)+\nabla f(x)\delta$, and then set the linearization to zero.
If we update the value of $x$ to be $x\gets x+\delta$, this gives the Newton--Raphson method.

[GENERAL DISCUSSION]

See also [[As a first approximation, neglect lower order terms]].

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Use an approximation as if it were exact

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