approximation

(noun)

An imprecise solution or result that is adequate for a defined purpose.

Related Terms

Examples of approximation in the following topics:

  • Linear Approximation

    • A linear approximation is an approximation of a general function using a linear function.
    • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
    • Linear approximations are widely used to solve (or approximate solutions to) equations.
    • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
    • If $f$ is concave-up, the approximation will be an underestimate.

  • Approximate Integration

    • Here, we will study a very simple approximation technique, called a trapezoidal rule.
    • The trapezoidal rule works by approximating the region under the graph of the function $f(x)$ as a trapezoid and calculating its area.
    • Although the method can adopt a nonuniform grid as well, this example used a uniform grid for the the approximation.
    • The function $f(x)$ (in blue) is approximated by a linear function (in red).
    • Use the trapezoidal rule to approximate the value of a definite integral

  • Newton's Method

    • Newton's Method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is x0 - f(x0) / f'(x0).
    • This $x$-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
    • We see that $x_{n+1}$ is a better approximation than $x_n$ for the root $x$ of the function $f$.

  • Tangent Planes and Linear Approximations

    • It is the best approximation of the surface by a plane at $p$, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to $p$ as these points converge to $p$.
    • Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point.
    • The approximation works well as long as the point $(x,y,z) $ under consideration is close enough to $(x_0,y_0,z_0)$, where the tangent plane touches the surface.
    • The plane describing the linear approximation for a surface described by $z=f(x,y)$ is given as:
    • Explain why the tangent plane can be used to approximate the surface near the point

  • Numerical Integration

    • Numerical integration is a method of approximating the value of a definite integral.
    • The integration points and weights depend on the specific method used and the accuracy required from the approximation.
    • The area can then be approximated by adding up the areas of the rectangles.
    • Notice that the smaller the rectangles are made, the more accurate the approximation.
    • For either one of these rules, we can make a more accurate approximation by breaking up the interval $[a, b]$ into some number $n$ of subintervals, computing an approximation for each subinterval, then adding up all the results.

  • Area and Distances

    • Decreasing the width of the approximation rectangles should yield a better result, so we will cross the interval in five steps, using the approximation points $0$, $\frac{1}{5}$, $\frac{2}{5}$, and so on, up to $1$.
    • Summing the areas of these rectangles, we get a better approximation for the sought integral, namely:
    • We can easily see that the approximation is still too large.
    • Using more steps produces a closer approximation, but will never be exact: replacing the $5$ subintervals by twelve as depicted, we will get an approximate value for the area of $0.6203$, which is too small.
    • For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean theorem.

  • The Definite Integral

    • As a first approximation, look at the unit square given by the sides $x = 0$ to $x = 1$, $y = f(0) = 0$, and $y = f(1) = 1$.
    • Decreasing the width of the approximation rectangles should yield a better result, so we will cross the interval in five steps, using the approximation points $0$, $\frac{1}{5}$, $\frac{2}{5}$, and so on, up to $1$.
    • Summing the areas of these rectangles, we get a better approximation for the sought integral, namely:
    • We can easily see that the approximation is still too large.
    • Using more steps produces a closer approximation, but will never be exact: replacing the $5$ subintervals by twelve as depicted, we will get an approximate value for the area of $0.6203$, which is too small.

  • Applications of Taylor Series

    • Taylor series expansion can help approximating values of functions and evaluating definite integrals.
    • The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function.
    • These approximations are often good enough if sufficiently many terms are included.
    • Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.
    • This image shows $\sin x$ and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

  • The Natural Logarithmic Function: Differentiation and Integration

    • The natural logarithm, generally written as $\ln(x)$, is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to $2.718281828$.
    • The Taylor polynomials for $\ln(1 + x)$ only provide accurate approximations in the range $-1 < x \leq 1$.
    • Note that, for $x>1$, the Taylor polynomials of higher degree are worse approximations.

  • Numerical Integration

    • The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:
    • If $f(x)$ is a smooth well-behaved function, integrated over a small number of dimensions and the limits of integration are bounded, there are many methods of approximating the integral with arbitrary precision.
    • It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative.
    • For either one of these rules, we can make a more accurate approximation by breaking up the interval $[a, b]$ into some number $n$ of subintervals, computing an approximation for each subinterval, then adding up all the results.