Numerical analysis

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Babylonian clay tablet YBC 7289 (c. 1800-1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 + 10/60 = 1.41421296... Ybc7289-bw.jpg
Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 + 10/60 = 1.41421296...

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Algorithm an unambiguous specification of how to solve a class of problems

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, and other tasks.

An approximation is anything that is similar but not exactly equal to something else.

Mathematical analysis branch of pure mathematics

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Contents

Before the advent of modern computers, numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

Interpolation method for constructing new data from known data

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

Differential equation mathematical equation that contains derivatives of an unknown function

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction. [2]

Yale Babylonian Collection

Comprising some 45,000 items, the Yale Babylonian Collection is an independent branch of the Yale University Library housed on the Yale University campus in Sterling Memorial Library at New Haven, Connecticut, United States.

YBC 7289

YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". It is believed to be the work of a student in southern Mesopotamia from some time in the range from 1800–1600 BC, and was donated to the Yale Babylonian Collection by J. P. Morgan.

Sexagesimal

Sexagesimal is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of the square root of 2, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Babylonian mathematics mathematics

Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time they fall in two distinct groups: one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia.

General introduction

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions

Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs.

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

Partial differential equation differential equation that contains unknown multivariable functions and their partial derivatives

In mathematics, a partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

The rest of this section outlines several important themes of numerical analysis.

History

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

Linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

In numerical analysis, Newton's method, named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. It is one example of a root-finding algorithm.

Lagrange polynomial

In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points with no two values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value the corresponding value . The interpolating polynomial of the least degree is unique, however, and since it can be arrived at through multiple methods, referring to "the Lagrange polynomial" is perhaps not as correct as referring to "the Lagrange form" of that unique polynomial.

To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.

The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

Direct and iterative methods

Direct vs iterative methods

Consider the problem of solving

3x3 + 4 = 28

for the unknown quantity x.

Direct method
3x3 + 4 = 28.
Subtract 43x3 = 24.
Divide by 3x3 =  8.
Take cube rootsx =  2.

For the iterative method, apply the bisection method to f(x) = 3x3 24. The initial values are a = 0, b = 3, f(a) = 24, f(b) = 57.

Iterative method
abmidf(mid)
031.513.875
1.532.2510.17...
1.52.251.8754.22...
1.8752.252.06252.32...

We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.

Discretization and numerical integration

Schumacher (Ferrari) in practice at USGP 2005.jpg

In a two-hour race, we have measured the speed of the car at three instants and recorded them in the following table.

Time0:201:001:40
km/h140150180

A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3 h × 140 km/h) = 93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity.

Ill-conditioned problem: Take the function f(x) = 1/(x  1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem.

Well-conditioned problem: By contrast, evaluating the same function f(x) = 1/(x  1) near x = 10 is a well-conditioned problem. For instance, f(10) = 1/9 ≈ 0.111 and f(11) = 0.1: a modest change in x leads to a modest change in f(x).

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability).

In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

Discretization

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization . For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

Generation and propagation of errors

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.

Round-off

Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).

Truncation and discretization error

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of , after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type is even more inexact.

What does it mean when we say that the truncation error is created when we approximate a mathematical procedure? We know that to integrate a function exactly requires one to find the sum of infinite trapezoids. But numerically one can find the sum of only finite trapezoids, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches zero but numerically we can only choose a finite value of the differential element.

Numerical stability and well-posed problems

Numerical stability is an important notion in numerical analysis. An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned , meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is ill-conditioned, then any small error in the data will grow to be a large error.

Both the original problem and the algorithm used to solve that problem can be well-conditioned and/or ill-conditioned, and any combination is possible.

So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x0 to , for instance x0 = 1.4, and then computing improved guesses x1, x2, etc. One such method is the famous Babylonian method, which is given by xk+1 = xk/2 + 1/xk. Another method, which we will call Method X, is given by xk+1 = (xk2 − 2)2 + xk. [3] We have calculated a few iterations of each scheme in table form below, with initial guesses x0 = 1.4 and x0 = 1.42.

BabylonianBabylonianMethod XMethod X
x0 = 1.4x0 = 1.42x0 = 1.4x0 = 1.42
x1 = 1.4142857...x1 = 1.41422535...x1 = 1.4016x1 = 1.42026896
x2 = 1.414213564...x2 = 1.41421356242...x2 = 1.4028614...x2 = 1.42056...
......
x1000000 = 1.41421...x27 = 7280.2284...

Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess x0 = 1.4 and diverges for initial guess x0 = 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

Numerical stability is affected by the number of the significant digits the machine keeps on, if we use a machine that keeps only the four most significant decimal digits, a good example on loss of significance is given by these two equivalent functions
If we compare the results of
and
by looking to the two results above, we realize that loss of significance (caused here by catastrophic cancelation) has a huge effect on the results, even though both functions are equivalent, as shown below
The desired value, computed using infinite precision, is 11.174755...

Areas of study

The field of numerical analysis includes many sub-disciplines. Some of the major ones are:

Computing values of functions

Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm.

Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year.

A line through 20 points Linear-regression.svg
A line through 20 points

Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points.

How much for a glass of lemonade? LemonadeJuly2006.JPG
How much for a glass of lemonade?

Optimization: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one glass of lemonade less per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day.

Wind direction in blue, true trajectory in black, Euler method in red. Wind-particle.png
Wind direction in blue, true trajectory in black, Euler method in red.

Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.

Interpolation, extrapolation, and regression

Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?

Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.

Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this.

Solving equations and systems of equations

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation is linear while is not.

Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.

Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.

Solving eigenvalue or singular value problems

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm [4] is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.

Optimization

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.

The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.

Evaluating integrals

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.

Differential equations

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.

Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

Software

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the GNU Scientific Library.

There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, and IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS) and Python with libraries such as NumPy, SciPy and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude. [5] [6]

Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.

Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.

See also

Notes

  1. Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
  2. The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a building
  3. This is a fixed point iteration for the equation , whose solutions include . The iterates always move to the right since . Hence converges and diverges.
  4. The Singular Value Decomposition and Its Applications in Image Compression Archived 4 October 2006 at the Wayback Machine
  5. Speed comparison of various number crunching packages Archived 5 October 2006 at the Wayback Machine
  6. Comparison of mathematical programs for data analysis Stefan Steinhaus, ScientificWeb.com

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In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.

Numerical methods for ordinary differential equations

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Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis, and a type of linear algebra. Because computers use floating-point arithmetic, they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision.

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Finite element method Numerical method for solving physical or engineering problems

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Parareal is a parallel algorithm from numerical analysis and used for the solution of initial value problems. It has been introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of the most widely studied parallel-in-time integration methods.

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