An Introduction to Numerical Analysis - Solutions
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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.
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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. 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. 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. 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 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 errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution.
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Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. Therefore, there is a truncation error of 0. Once an error is generated, it will generally propagate through the calculation. The truncation error is created when a mathematical procedure is approximated.
To integrate a function exactly it is required to find the sum of infinite trapezoids, but numerically only the sum of only finite trapezoids can be found, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches zero but numerically only a finite value of the differential element can be chosen. Numerical stability is a 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.
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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' 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.
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Hence, the Babylonian method is numerically stable, while Method X is numerically unstable. Interpolation: Observing that the temperature varies from 20 degrees Celsius at to 14 degrees at , a linear interpolation of this data would conclude that it was 17 degrees at and Regression: In linear regression, given n points, a line is computed that passes as close as possible to those n points.
Differential equation: If fans are set up to blow air from one end of the room to the other and then a feather isdropped 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 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 the value of the unknown function at a point which is outside the given points must be found. 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 , the unknown function can be found.
The least squares -method is one way to achieve this. 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. Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i. 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. Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions.
An Introduction to Numerical Analysis: Solutions Manual - Kendall E. Atkinson - Google книги
For instance, the spectral image compression algorithm  is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis. 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. More questions? Visit the Learner Help Center. Browse Chevron Right. Physical Science and Engineering Chevron Right.
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