Numerical Analysis

Numerical Analysis deals with the process of getting the numerical solution to complex problems. The majority of mathematical problems in science and engineering are difficult to answer precisely, and in some cases it is impossible. To make a tough Mathematical problem easier to solve, an approximation is essential. Numerical approximation has become more popular as a result of tremendous advances in computational technology. As a result, a great deal of scientific software is being developed to solve more complex challenges quickly and easily. Let us go through the definition of numerical analysis as well as the various concepts included, such as errors, interpolation and so on in this article.

Introduction to Numerical Analysis

Numerical analysis is a discipline of mathematics concerned with the development of efficient methods for getting numerical solutions to complex mathematical problems. There are three sections to the numerical analysis. The first section of the subject deals with the creation of a problem-solving approach. The analysis of methods, which includes error analysis and efficiency analysis, is covered in the second section. The efficiency analysis shows us how fast we can compute the result, while the error analysis informs us how correct the result will be if we utilize the approach. The construction of an efficient algorithm to implement the approach as a computer code is the subject’s third part. All three elements must be familiar to have a thorough understanding of the numerical analysis.

Meanwhile, there are at least three reasons to learn the theoretical foundations of numerical methods:

  • Learning various numerical methods and analyzing them will familiarize a person with the process of inventing new numerical methods. When the existing approaches are insufficient or inefficient to handle a certain problem, this is critical.
  • In many cases, there are multiple solutions to a problem. As a result, using the right procedure is critical for getting a precise answer in less time.
  • With a solid foundation, one can effectively apply methods (especially when a technique has its own restrictions and/or drawbacks in certain instances) and, more significantly, analyze what went wrong when results did not meet expectations.

Let’s have a look at some of the key topics in numerical analysis.

Different Types of Errors

The disparity between the approximate representation of a real number and the actual value is termed an error .

Error = True Value – Approximate Value , is the formula for calculating the error in a computed amount.

The absolute error is defined as the absolute value of the error defined above.

Relative Error = Error / True Value is a measurement of the error in respect to the magnitude of the true value.

The relative error is multiplied by 100 to get the percentage error .

The phrase “ truncation error ” refers to the error that occurs when a smooth function is approximated by reducing its Taylor series representation to a limited number of terms.

Significant Digits

If x A is an approximation to x, so we can conclude that x A approximates x to r significant β-digits if |x − x A | ≤ (½)β s−r+1 with “s” the greatest integer such that β s ≤ |x|.

As an example, the approximate value x A = 0.333 includes three significant digits for x = ⅓, since |x − x A | ≈ .00033 < 0.0005 = 0.5 × 10 −3 .

But 10−1 < 0.333 · · · = x.

Hence, in this case s = −1 and and therefore r = 3.

Propagation of Errors

When an error is committed, it has an impact on subsequent outcomes because it propagates through subsequent calculations. We’ll look at how utilizing approximate numbers rather than actual numbers affects the outcomes before moving on to function evaluation. We’ll now explore how error propagates in four basic arithmetic operations .

  • In addition and subtraction, the total of the error bounds for the terms provides an error bound for the results .
  • In multiplication and division, The sum of the bounds for the relative errors of the given integers gives a limitation for the relative error of the results.

Finite Difference Operators

Now, let us discuss the various finite difference operators in brief.

Forward Operator

Assume that “h” be the finite difference, then

Δf(x) = f(x+h) – f(x)

Δ 2 f(x) = f(x+2h)-2f(x+h) + f(x)

Δ 3 f(x)= f(x+3h) – 3f(x+2h) + 2f(x+h) – f(x)

Shift Operator

Assume that h be the finite difference.

Then, E f(x) = f(x+h)

E n f(x) = f(x+nh)

Backward Difference

Suppose h be the finite difference.

Central Difference Operator

Averaging operator, factorial notation, relation between different finite operators.

Relationship Between Δ and E

E ≡ 1 + Δ and Δ ≡ E-1

Hence, E n ≡ (1+Δ) n and Δ n ≡ (E-1) n

Interpolation

Interpolation is the process of determining the approximate value of a function f(x) for an x between multiple x values x 0, x 1 , …, x n for which the value of f(x) is known.

I.e., f(x i ) = f i (i = 0, 1, 2, …, n)

If the real-valued function f(x) has (n+1) different values, then x 0 x 1 , ..x n . A polynomial of degree n or less is P n (x i ) = f(x). It indicates that there can only be one polynomial with a degree less than or equal to n that interpolates f(x) at (n+1) unique points x 0 , x 1 , x 2 , …x n .

Solved Example on Numerical Analysis

Show that μ 4 = μ 3 + Δμ 2 + Δ 2 μ 1 + Δ 3 μ 1

As we know that

Δμ x = μ x+h – μ x

Hence, μ 4 – μ 3 = Δμ 3

μ 3 – μ 2 = Δμ 2

μ 2 – μ 1 = Δμ 1

μ 4 = μ 3 + Δμ 3

μ 4 = μ 3 + Δμ 2 – Δ 2 μ 2

μ 4 = μ 3 + Δμ 2 + Δ 2 μ 1 + Δ 3 μ 1

Hence, proved.

Stay tuned to BYJU’S – The Learning App and download the app all the Maths-related concepts easily by exploring more videos.

Frequently Asked Questions on Numerical Analysis

What is numerical analysis.

Numerical analysis is a branch of mathematics concerned with the development of efficient methods for solving complicated mathematical problems numerically.

What are the different types of numerical analysis?

The different types of numerical analysis are finite difference methods, propagation of errors, interpolation methods, and so on.

Is calculus required for learning numerical analysis?

Yes, calculus is required for learning numerical analysis, as we should know differential integration.

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Numerical Analysis

The field of numerical analysis focuses on algorithms that use numerical approximation for the problems of mathematical analysis. Wolfram|Alpha provides algorithms for solving integrals, differential equations and the roots of equations through a variety of numerical methods. Compare different methods for accuracy and speed. Use fine control over such parameters as step size or starting point.

Compute roots using specific starting points, precisions and numerical methods.

Find the roots of an equation using Newton's method:

Find the roots of an equation using the secant method:, compute the n th root of a number using the bisection method:.

Compute solutions to ordinary differential equations using numerical methods, such as Euler's method, the midpoint method and the Runge–Kutta methods.

Solve an ODE using a specified numerical method:

Specify an adaptive method:.

Use numerical integration methods, such as the trapezoidal rule, to solve integrals.

Numerically integrate functions that cannot be integrated symbolically:

Approximate an integral using a specified numerical method:, related examples.

  • Calculus & Analysis
  • Differential Equations

Numerical analysis

Curator: Kendall E. Atkinson

Eugene M. Izhikevich

WikiSysop Real Name

Lawrence F. Shampine

James Meiss

Skip Thompson

Dr. Kendall E. Atkinson , Department of Computer Science, Department of Mathematics, University of Iowa

Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics.  Such problems originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. These problems occur throughout the natural sciences, social sciences, medicine, engineering, and business. Beginning in the 1940's, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science, medicine, engineering, and business; and numerical analysis of increasing sophistication has been needed to solve these more accurate and complex mathematical models of the world. The formal academic area of numerical analysis varies from highly theoretical mathematical studies to computer science issues involving the effects of computer hardware and software on the implementation of specific algorithms.

Areas of numerical analysis

A rough categorization of the principal areas of numerical analysis is given below, keeping in mind that there is often a great deal of overlap between the listed areas. In addition, the numerical solution of many mathematical problems involves some combination of some of these areas, possibly all of them. There are also a few problems which do not fit neatly into any of the following categories.

Systems of linear and nonlinear equations

  • Numerical solution of systems of linear equations . This refers to solving for \(x\) in the equation \(Ax=b\) with given matrix \(A\) and column vector \(b\ .\) The most important case has \(A\) a square matrix. There are both direct methods of solution (requiring only a finite number of arithmetic operations) and iterative methods (giving increased accuracy with each new iteration). This topic also includes the matrix eigenvalue problem for a square matrix \(A\ ,\) solving for \(x\) and \(\lambda\) in the equation \(Ax=\lambda x\ .\)
  • Numerical solution of systems of nonlinear equations . This refers to rootfinding problems which are usually written as \(f(x)=0\) with \(x\) a vector with \(n\) components and \(f(x)\) a vector with \(m\) components. The most important case has \(n=m\ .\)
  • Optimization . This refers to minimizing or maximizing a real-valued function \(f(x)\ .\) The permitted values for \(x=(x_{1},\dots,x_{n}) \) can be either constrained or unconstrained. The 'linear programming problem' is a well-known and important case; \(f(x)\) is linear, and there are linear equality and/or inequality constraints on \(x\ .\)

Approximation theory

Use computable functions \(p(x)\) to approximate the values of functions \(f(x)\) that are not easily computable or use approximations to simplify dealing with such functions. The most popular types of computable functions \(p(x)\) are polynomials, rational functions, and piecewise versions of them, for example spline functions. Trigonometric polynomials are also a very useful choice.

  • Best approximations . Here a given function \(f(x)\) is approximated within a given finite-dimensional family of computable functions. The quality of the approximation is expressed by a functional, usually the maximum absolute value of the approximation error or an integral involving the error. Least squares approximations and minimax approximations are the most popular choices.
  • Interpolation . A computable function \(p(x)\) is to be chosen to agree with a given \(f(x)\) at a given finite set of points \(x\ .\) The study of determining and analyzing such interpolation functions is still an active area of research, particularly when \(p(x)\) is a multivariate polynomial.
  • Fourier series . A function \(f(x)\) is decomposed into orthogonal components based on a given orthogonal basis \(\left\{ \varphi _{1},\varphi_{2},\dots\right\} \ ,\) and then \(f(x)\) is approximated by using only the largest of such components. The convergence of Fourier series is a classical area of mathematics, and it is very important in many fields of application. The development of the Fast Fourier Transform in 1965 spawned a rapid progress in digital technology. In the 1990s wavelets became an important tool in this area.
  • Numerical integration and differentiation . Most integrals cannot be evaluated directly in terms of elementary functions, and instead they must be approximated numerically. Most functions can be differentiated analytically, but there is still a need for numerical differentiation, both to approximate the derivative of numerical data and to obtain approximations for discretizing differential equations.

Numerical solution of differential and integral equations

These equations occur widely as mathematical models for the physical world, and their numerical solution is important throughout the sciences and engineering.

  • Ordinary differential equations . This refers to systems of differential equations in which the unknown solutions are functions of only a single variable. The most important cases are initial value problems and boundary value problems , and these are the subjects of a number of textbooks. Of more recent interest are ' differential-algebraic equations ', which are mixed systems of algebraic equations and ordinary differential equations. Also of recent interest are ' delay differential equations ', in which the rate of change of the solution depends on the state of the system at past times.
  • Partial differential equations . This refers to differential equations in which the unknown solution is a function of more than one variable. These equations occur in almost all areas of engineering, and many basic models of the physical sciences are given as partial differential equations. Thus such equations are a very important topic for numerical analysis. For example, the Navier-Stokes equations are the main theoretical model for the motion of fluids, and the very large area of 'computational fluid mechanics' is concerned with solving numerically these and other equations of fluid dynamics
  • Integral equations . These equations involve the integration of an unknown function, and linear equations probably occur most frequently. Some mathematical models lead directly to integral equations; for example, the radiosity equation is a model for radiative heat transfer. Another important source of such problems is the reformulation of partial differential equations, and such reformulations are often called 'boundary integral equations'.

Some common viewpoints and concerns in numerical analysis

Most numerical analysts specialize in small sub-areas of the areas listed above, but they share some common concerns and perspectives. These include the following.

  • The mathematical aspects of numerical analysis make use of the language and results of linear algebra, real analysis, and functional analysis.
  • If you cannot solve a problem directly, then replace it with a 'nearby problem' which can be solved more easily. This is an important perspective which cuts across all types of mathematical problems. For example, to evaluate a definite integral numerically, begin by approximating its integrand using polynomial interpolation or a Taylor series, and then integrate exactly the polynomial approximation.
  • All numerical calculations are carried out using finite precision arithmetic, usually in a framework of floating-point representation of numbers. What are the effects of using such finite precision computer arithmetic? How are arithmetic calculations to be carried out? Using finite precision arithmetic will affect how we compute solutions to all types of problems, and it forces us to think about the limits on the accuracy with which a problem can be solved numerically. Even when solving finite systems of linear equations by direct numerical methods, infinite precision arithmetic is needed in order to find a particular exact solution.
  • There is concern with ' stability ', a concept referring to the sensitivity of the solution of a given problem to small changes in the data or the given parameters of the problem. There are two aspects to this. First, how sensitive is the original problem to small changes in the data of the problem? Problems that are very sensitive are often referred to as 'ill-conditioned' or 'ill-posed', depending on the degree of sensitivity. Second, the numerical method should not introduce additional sensitivity that is not present in the original mathematical problem being solved. In developing a numerical method to solve a problem, the method should be no more sensitive to changes in the data than is true of the original mathematical problem.
  • There is a fundamental concern with error, its size, and its analytic form. When approximating a problem, a numerical analyst would want to understand the behaviour of the error in the computed solution. Understanding the form of the error may allow one to minimize or estimate it. A 'forward error analysis' looks at the effect of errors made in the solution process. This is the standard way of understanding the consequences of the approximation errors that occur in setting up a numerical method of solution, e.g. in numerical integration and in the numerical solution of differential and integral equations. A 'backward error analysis' works backward in a numerical algorithm, showing that the approximating numerical solution is the exact solution to a perturbed version of the original mathematical problem. In this way the stability of the original problem can be used to explain possible difficulties in a numerical method. Backward error analysis has been especially important in understanding the behaviour of numerical methods for solving linear algebra problems.
  • In order to develop efficient means of calculating a numerical solution, it is important to understand the characteristics of the computer being used. For example, the structure of the computer memory is often very important in devising efficient algorithms for large linear algebra problems. Also, parallel computer architectures lead to efficient algorithms only if the algorithm is designed to take advantage of the parallelism.
  • Numerical analysts are generally interested in measuring the efficiency of algorithms. What is the cost of a particular algorithm? For example, the use of Gaussian elimination to solve a linear system \(Ax=b\) containing \(n\) equations will require approximately \(2n^{3}/3\) arithmetic operations. How does this compare with other numerical methods for solving this problem? This topic is a part of the larger area of 'computational complexity '.
  • Use information gained in solving a problem to improve the solution procedure for that problem. Often we do not fully understand the characteristics of a problem, especially very complicated and large ones. Such a solution process is sometimes referred to as being an 'adaptive procedure', and it can also be viewed as a feedback process.

Development of numerical methods

Numerical analysts and applied mathematicians have a variety of tools which they use in developing numerical methods for solving mathematical problems. An important perspective, one mentioned earlier, which cuts across all types of mathematical problems is that of replacing the given problem with a 'nearby problem' which can be solved more easily. There are other perspectives which vary with the type of mathematical problem being solved.

Numerical solution of systems of linear equations

Linear systems arise in many of the problems of numerical analysis, a reflection of the approximation of mathematical problems using linearization. This leads to diversity in the characteristics of linear systems, and for this reason there are numerous approaches to solving linear systems. As an example, numerical methods for solving partial differential equations often lead to very large 'sparse' linear systems in which most coefficients are zero. Solving such sparse systems requires methods that are quite different from those used to solve more moderate sized 'dense' linear systems in which most coefficients are non-zero.

There are 'direct methods' and 'iterative methods' for solving all types of linear systems, and the method of choice depends on the characteristics of both the linear system and on the computer hardware being used. For example, some sparse systems can be solved by direct methods, whereas others are better solved using iteration. With iteration methods, the linear system is sometimes transformed to an equivalent form that is more amenable to being solved by iteration; this is often called 'pre-conditioning' of the linear system.

With the matrix eigenvalue problem \(Ax=\lambda x\ ,\) it is standard to transform the matrix \(A\) to a simpler form, one for which the eigenvalue problem can be solved more easily and/or cheaply. A favorite choice are 'orthogonal transformations' because they are a simple and stable way to convert the given matrix \(A\ .\) Orthogonal transformations are also very useful in transforming other problems in numerical linear algebra . Of particular importance in this regard is the least squares solution of over-determined linear systems.

The linear programming problem was solved principally by the 'simplex method' until new approaches were developed in the 1980s, and it remains an important method of solution. The simplex method is a direct method that uses tools from the numerical solution of linear systems.

Numerical solution of systems of nonlinear equations

With a single equation \(f(x)=0\ ,\) and having an initial estimate \(x_{0}\) of the root \(\alpha\ ,\) approximate \(f(x)\) by its tangent line at the point \(\left(x_{0},f(x_{0})\right) \ .\) Find the root of this tangent line as an approximation to the root \(\alpha\) of the original equation \(f(x)=0\ .\) This leads to 'Newton's iteration method', \[ x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})},\quad\quad n=0,1,\dots \] Other linear and higher degree approximations can be used, and these lead to alternative iteration methods. An important derivative-free approximation of Newton’s method is the 'secant method'.

For a system of \(m\) nonlinear equations for a solution vector \(x\) in \(R^m\ ,\) we approximate \(f(x)\) by its linear Taylor approximation about the initial estimate \(x_{0}\ .\) This leads to Newton's method for nonlinear systems, \[ x_{n+1}=x_{n}-[f^{\prime}(x_{n})]^{-1}f(x_{n}),\quad\quad n=0,1,\dots \] in which \(f^{\prime}(x)\) denotes the Jacobian matrix , of order \(m\times m\) for \(f(x)\ .\)

In practice, the Jacobian matrix for \(f(x)\) is often too complicated to compute directly; instead the partial derivatives in the Jacobian matrix are approximated using 'finite differences'. This leads to a 'finite difference Newton method'. As an alternative strategy and in analogy with the development of the secant method for the single variable problem, there is a similar rootfinding iteration method for solving nonlinear systems. It is called 'Broyden’s method' and it uses finite difference approximations of the derivatives in the Jacobian matrix, avoiding the evaluation of the partial derivatives of \(f(x)\ .\)

Numerical methods for solving differential and integral equations

With such equations, there are usually at least two general steps involved in obtaining a nearby problem from which a numerical approximation can be computed; this is often referred to as 'discretization' of the original problem. The given equation will have a domain on which the unknown function is defined, perhaps an interval in one dimension and maybe a rectangle, ellipse, or other simply connected bounded region in two dimensions. Many numerical methods begin by introducing a mesh or grid on this domain, and the solution is to be approximated using this grid. Following this, there are several common approaches.

One approach approximates the equation with a simpler equation defined on the mesh. For example, consider approximating the boundary value problem \[ u^{\prime\prime}(s)=f\left( s,u(s)\right) ,\quad0\leq s\leq1 \] \[ u(0)=u(1)=0. \] Introduce a set of mesh points \(s_{j}=jh\ ,\) \(j=0,1,\dots,n\ ,\) with \(h=1/n\) for some given \(n\geq2\ .\) Approximate the boundary value problem by \[ \frac{1}{h^{2}}\left[ \tilde{u}_{n}(s_{j+1})-2\tilde{u}_{n}(s_{j})+\tilde {u}_{n}(s_{j-1})\right] =f\left( s_{j},\tilde{u}_{n}\left( s_{j}\right) \right) ,\quad j=1,\dots,n-1 \] \[ \tilde{u}_{n}(0)=\tilde{u}_{n}(1)=0 \] The second derivative in the original problem has been replaced by a numerical approximation to the second derivative. The new problem is a finite system of nonlinear equations, presumably amenable to solution by known techniques. The solution to this new problem is \(\tilde{u}_{n}\ ,\) and it is defined on only the mesh points \(\left\{ s_{j}\right\} \ .\)

A second approach to discretizing differential and integral equations is as follows. Choose a finite-dimensional family of functions, denoted here by \(\mathcal{F}\ ,\) with which to approximate the unknown solution function \(u\ .\) Write the given differential or integral equation as \(L\left( u\right) =0\ ,\) with \(L(v)\) a function for any function \(v\ ,\) perhaps over a restricted class of functions \(v\ .\) The numerical method consists of selecting a function \(\tilde{u}\in\mathcal{F}\) such that \(L(\tilde{u})\) is a small function in some sense. The various ways of doing this lead to ' Galerkin methods ', 'collocation methods', and 'least square methods'.

Yet another approach is to reformulate the equation \(L\left( u\right) =0\) as an optimization problem. Such reformulations are a part of the classical area of mathematics known as the 'calculus of variations', a subject that reflects the importance in physics of minimization principles. The well-known ' finite element method ' for solving elliptic partial differential equations is obtained in this way, although it often coincides with a Galerkin method.

The approximating functions in \(\mathcal{F}\) are often chosen as piecewise polynomial functions which are polynomial over the elements of the mesh chosen earlier. Such methods are sometimes called 'local methods'. When the approximating functions \(p\in\mathcal{F}\) are defined without reference to a grid, then the methods are sometimes called 'global methods' or 'spectral methods'. Examples of such \(\mathcal{F}\) are sets of polynomials or trigonometric functions of some finite degree or less.

With all three approaches to solving a differential or integral equations, the intent is that the resulting solution \(\tilde{u}\) be close to the desired solution \(u\ .\) The business of theoretical numerical analysis is to analyze such an algorithm and investigate the size of \(u-\tilde{u}\ .\)

For an historical account of early numerical analysis, see

  • Herman Goldstine. A History of Numerical Analysis From the 16th Through the19th Century , Springer-Verlag, New York, 1977.

For a current view of numerical analysis as taught at the undergraduate level, see

  • Cleve Moler. Numerical Computing with MATLAB , SIAM Pub., Philadelphia, 2004.

For a current view of numerical analysis as taught at the advanced undergraduate or beginning graduate level, see

  • Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics , Springer-Verlag, New York, 2000.
  • Christoph W. Ueberhuber. Numerical Computation: Vol. 1: Methods, Software, and Analysis, Vol. 2: Methods, Software, and Analysis , Springer-Verlag, New York, 1997.

For one perspective on a theoretical framework using functional analysis for studying many problems in numerical analysis, see

  • Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework , 2nd ed., Springer-Verlag, New York, 2005.

As references for numerical linear algebra, see

  • Gene Golub and Charles Van Loan. Matrix Computations , 3rd ed., Johns Hopkins University Press, 1996.
  • Nicholas Higham. Accuracy and Stability of Numerical Algorithms , SIAM Pub., Philadelphia, 1996.

For an introduction to practical numerical analysis for solving ordinary differential equations, see

  • Lawrence Shampine, Ian Gladwell, Skip Thompson. Solving ODEs with Matlab , Cambridge University Press, Cambridge, 2003.

For information on computing aspects of numerical analysis, see

  • Michael Overton. Numerical computing with IEEE floating point arithmetic , SIAM Pub., Philadelphia, 2001.
  • Suely Oliveira and David Stewart. Writing Scientific Software: A Guide to Good Style , Cambridge University Press, Cambridge, 2006.

Internal references

  • Olaf Sporns (2007) Complexity . Scholarpedia , 2(10):1623.
  • Skip Thompson (2007) Delay-differential equations . Scholarpedia, 2(3):2367.
  • James Meiss (2007) Dynamical systems . Scholarpedia, 2(2):1629.
  • Eugene M. Izhikevich (2007) Equilibrium . Scholarpedia, 2(10):2014.
  • Lawrence F. Shampine and Skip Thompson (2007) Initial value problems . Scholarpedia, 2(3):2861.
  • Mark Aronoff (2007) Language . Scholarpedia, 2(5):3175.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability . Scholarpedia, 1(10):1838.
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  • Numerical Analysis

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An Introduction to Numerical Analysis

Numerical Analysis is the Mathematics branch responsible for designing effective ways to find numerical solutions to complex Mathematical problems. Most Mathematical problems from science and engineering are very complex and sometimes cannot be solved directly. Therefore, measuring a complex Mathematical problem is very important to make it easier to solve. Due to the great advances in computational technology, numeracy has become very popular and is a modern tool for scientists and engineers. As a result many software programs are being developed such as Matlab, Mathematica, Maple etc. the most difficult problems in an effective and simple way. These softwares contain functions that use standard numeric methods, in which the user can bypass the required parameters and obtain the results in a single command without knowing the numerical details.

The Numerical Analysis method is mainly used in the area of Mathematics and Computer Science that creates, analyzes, and implements algorithms for solving numerical problems of continuous Mathematics. Such types of problems generally originate from real-world applications of algebra, geometry and calculus, and they also involve variables that vary continuously. These problems occur throughout the natural sciences, social sciences, engineering, medicine, and the field of business. Introduction of Numerical Analysis during the past half-century, the growth in power and availability of digital computers has led to the increasing use of realistic Mathematical models in science and engineering. Here we will learn more about numerical method and analysis of numerical methods.

Numerical Method

Numerical methods are techniques that are used to approximate Mathematical procedures.  We need approximations because we either cannot solve the procedure analytically or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns). 

Different Types of Numerical Methods

The numerical analysts and Mathematicians used have a variety of tools that they use to develop numerical methods for solving Mathematical problems. The most important idea, mentioned earlier, that cuts across all sorts of Mathematical problems is that of changing a given problem with a 'near problem' that can be easily solved. There are other ideas that differ on the type of Mathematical problem solved.

An Introduction to Numerical Methods for Solving Common Division Problems Given Below:

Euler method - the most basic way to solve ODE

Clear and vague methods - vague methods need to solve the problem in every step

The Euler Back Road - the obvious variation of the Euler method

Trapezoidal law - the direct method of the second system

Runge-Kutta Methods - one of the two main categories of problems of the first value .

Numerical Methods

Newton method

Some calculations cannot be solved using algebra or other Mathematical methods. For this we need to use numerical methods. Newton's method is one such method and allows us to calculate the solution of f (x) = 0.

Simpson Law

The other important ones cannot be assessed in terms of integration rules or basic functions. Simpson's law is a numerical method that calculates the numerical value of a direct combination.

Trapezoidal law

A trapezoidal rule is a numerical method that calculates the numerical value of a direct combination. The other important ones cannot be assessed in terms of integration rules or basic functions.

Numerical Computation

The term “numerical computations” means to use computers for solving problems involving real numbers. In this process of problem-solving, we can distinguish several more or less distinct phases. The first phase is formulation. While formulating a Mathematical model of a physical situation, scientists should take into account the fact that they expect to solve a problem on a computer. Therefore they will provide for specific objectives, proper input data, adequate checks, and for the type and amount of output. 

Once a problem has been formulated, then the numerical methods, together with preliminary error analysis, must be devised for solving the problem. A numerical method that can be used to solve a problem is called an algorithm. An algorithm is a complete and unambiguous set of procedures that are used to find the solution to a Mathematical problem. The selection or construction of appropriate algorithms is done with the help of Numerical Analysis. We have to decide on a specific algorithm or set of algorithms for solving the problem, numerical analysts should also consider all the sources of error that may affect the results. They should consider how much accuracy is required. To estimate the magnitude of the round-off and discretization errors, and determine an appropriate step size or the number of iterations required.

The programmer should transform the suggested algorithm into a set of unambiguous that is followed by step-by-step instructions to the computer.  The flow chart is the first step in this procedure. A flow chart is simply a set of procedures, that are usually written in logical block form, which the computer will follow. The complexity of the flow will depend upon the complexity of the problem and the amount of detail included. However, it should be possible for someone else other than the programmer to follow the flow of information from the chart. The flow chart is an effective aid to the programmer, they must translate its major functions into a program. And, at the same time, it is an effective means of communication to others who wish to understand what the program does. 

Numerical Computing Characteristics

Accuracy: Every numerical method introduces errors. It may be due to the use of the proper Mathematical process or due to accurate representation and change of numbers on the computer. 

Efficiency: Another consideration in choosing a numerical method for a Mathematical model solution efficiency Means the amount of effort required by both people and computers to use the method.

Numerical instability: Another problem presented by a numerical method is numerical instability. Errors included in the calculation, from any source, increase in different ways. In some cases, these errors are usually rapid, resulting in catastrophic results.

Numerical Computing Process

Construction of a Mathematical model.

Construction of an appropriate numerical system.

Implementation of a solution.

Verification of the solution.

Trapezoidal Law

In Mathematics, trapezoidal law, also known as trapezoid law or trapezium law, is the most important measure of direct equity in Numerical Analysis. Trapezoidal law is a coupling law used to calculate the area under a curve by dividing the curve into a small trapezoid. The combination of all the small trapezoid areas will provide space under the curve. Let's understand the trapezoidal law formula and its evidence using examples in future sections.

Numerical and Statistical Methods

Numerical methods, as said above, are techniques to approximate Mathematical procedures. On the other hand, statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from the given data. Thus we can say, statistical methods are Mathematical formulas, models, and techniques that are used in the statistical analysis of raw research data. The application of statistical methods extracts information from research data and provides different methods to assess the robustness of research outputs. Some common statistical tools and procedures are given below :

Descriptive

Mean (average)

Inferential

Linear regression analysis

Analysis of variance

Null hypothesis testing

Introduction to Finite Element Method

The various laws of physics related to space and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). If we have the vast majority of geometries and problems, these PDEs cannot be solved using analytical methods. Instead of that, we have created an approximation of the equations, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. Thus, the solution to the numerical model equations is, in turn, an approximation of the real solution to the PDEs. The finite element method is used to compute such approximations.

The finite element method is a numerical technique that is used for solving problems that are described by partial differential equations or can be formulated as functional minimization. A domain of interest is represented by the assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field. A continuous physical problem is transformed into a discretized finite element problem with the help of unknown nodal values. For a linear problem, a system of linear algebraic equations must be solved. We can recover values inside finite elements using the nodal values.

Two Features of the Fem are Mentioned below:

Piecewise approximation of physical fields on finite elements provides good precision even with simple approximating functions (i.e. increasing the number of elements we can achieve any precision).

Locality of approximation leads to sparse equation systems that are mainly used for a discretized problem. With the help of this, we can solve problems with a very large number of nodal unknowns.

Typical Classes of Engineering Problems That Can be Solved Using Fem are:

Structural mechanics

Heat transfer

Electromagnetics

Finite Element Method MATLAB

Finite element analysis is a computational method for analyzing the behaviour of physical products under loads and boundary conditions. A typical FEA workflow in MATLAB includes 

Importing or creating geometry.

Generating mesh.

Defining physics of the problem with the help of load, boundary and initial conditions.

Solving and visualizing results.

The design of experiments or optimization techniques can be used along with FEA to perform trade-off studies or to design an optimal product for specific applications.

MATLAB is Very Useful Software and is Very Easy to Apply Finite Element Analysis Using MATLAB. It Helps Us in Applying Fem in Several Ways:

Partial differential equations (PDEs) can be solved using the inbuilt Partial Differential Equation Toolbox.

In MATLAB, with the help of Statistics and Machine Learning Toolbox, we can apply the design of experiments and other statistics and machine learning techniques with finite element analysis.

Also, the optimization techniques can be applied to FEM simulations to come up with an optimum design with Optimization Toolbox.

Parallel Computing Toolbox speeds up the analysis by distributing multiple Finite element analysis simulations to run in parallel.

arrow-right

FAQs on Numerical Analysis

1. What's the Trapezoidal Rule?

Trapezoidal Rule is an integration rule, in Calculus, that evaluates the location beneath the curves via dividing the total location into smaller trapezoids in preference to using rectangles.

2. Why is the guideline named after a trapezoid?

The call trapezoidal is because whilst the location under the curve is evaluated, then the full vicinity is divided into small trapezoids rather than rectangles. Then we find the region of these small trapezoids in a definite c program language period.

3. What is the use of Numerical techniques?

Numerical strategies are used in Mathematics and computer technological know-how that creates, analyzes, and implements algorithms to acquire the numerical answers to problems using non-stop variables. Such troubles rise up in the course of the herbal sciences, social sciences, engineering, medicine, and also in commercial enterprise.

4.  What are the basics of the Finite detail method?

The finite element approach is a Mathematical method used to calculate approximate answers to differential equations. The intention of this method is to convert the differential equations into hard and fast linear equations that can then be solved by the computer in a routine manner.

5. What is the distinction between the Trapezoidal Rule and Riemann Sums rule?

In the Trapezoidal Rule, we use trapezoids to approximate the region under the curve while in Riemann sums we use rectangles to discover areas below the curve, in case of integration.

6. Define the Trapezoid Rule of Numerical Analysis.

The trapezoidal rule is used to find the exact value of a definite integral using a numerical method. This rule is based on the concept of the Newton-Cotes formula which states that we can find the exact value of the integral as the nth order polynomial.

7. What is the Use of Numerical Methods?

Numerical methods are used in Mathematics and Computer Science that creates, analyzes, and implements algorithms to obtain the numerical solutions to problems using continuous variables. Such

8. What are the Basics of the Finite Element Method?

The finite element method is a Mathematical procedure used to calculate approximate solutions to differential equations. The goal of this method is to transform the differential equations into a set of linear equations that can then be solved by the computer in a routine manner.

9. Why is the guideline named after a trapezoid?

10. What is the use of Numerical techniques?

11.  What are the basics of the Finite detail method?

12. What is the distinction between the Trapezoidal Rule and Riemann Sums rule?

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Mathematics LibreTexts

1.01: Introduction to Numerical Methods

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Lesson 1: Why Numerical Methods?

Learning objectives.

After successful completion of this lesson, you should be able to: 1) Enumerate the need for numerical methods.

Introduction

Numerical methods are techniques to approximate mathematical processes (examples of mathematical processes are integrals, differential equations, nonlinear equations).

Approximations are needed because

1) we cannot solve the procedure analytically, such as the standard normal cumulative distribution function

\[\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{x}e^{- t^{2}/2}{dt} \;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{1.1}) \nonumber\]

2) the analytical method is intractable, such as solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss (Figure \(\PageIndex{1.1}\)).

A wooden truss holding up a roof.

In the case of Equation (1), an exact solution is not available for \(\Phi(x)\) other than for \(x = 0\) and \(x \rightarrow \infty\) . For other values of \(x\) where an exact solution is not available, one may solve the problem by using approximate techniques such as the left-hand Reimann sum you were introduced to in the Integral Calculus course.

In the truss problem, one can solve \(1000\) simultaneous linear equations for \(1000\) unknowns without using a calculator. One can use fractions, long divisions, and long multiplications to get the exact answer. But just the thought of such a task is laborious. The task may seem less laborious if we are allowed to use a calculator, but it would still fall under the category of an intractable, if not an impossible, problem. So, we need to find a numerical technique and convert it into a computer program that solves a set of \(n\) equations and \(n\) unknowns.

Again, what are numerical methods? They are techniques to solve a mathematical problem approximately. As we go through the course, you will see that numerical methods let us find solutions close to the exact one, and we can quantify the approximate error associated with the answer. After all, what good is an approximation without quantifying how good the approximation is?

Audiovisual Lecture

Title: Why Do We Need Numerical Methods

Summary : This video is an introduction to why we need numerical methods.

Lesson 2: Steps of Solving an Engineering Problem

After successful completion of this lesson, you should be able to:

1) go through the stages (problem description, mathematical modeling, solving and implementation) of solving a particular physical problem.

Numerical methods are used by engineers and scientists to solve problems. However, numerical methods are just one step in solving an engineering problem. There are four steps for solving an engineering problem, as shown in Figure \(\PageIndex{2.1}\).

Flowchart where problem description leads to mathematical model, which leads to solution of the mathematical model, which leads to using the solution.

The first step is to describe the problem. The description would involve writing the background of the problem and the need for its solution. The second step is developing a mathematical model for the problem, and this could include the use of experiments or/and theory. The third step involves solving the mathematical model. The solution may consist of analytical or/and numerical means. The fourth step is implementing the solution to see if the problem is solved.

Let us see through an example of these four steps of solving an engineering problem.

Problem Description

To make the fulcrum (Figure \(\PageIndex{2.2}\)) of a bascule bridge, a long hollow steel shaft called the trunnion is shrunk-fit into a steel hub. The resulting steel trunnion-hub assembly is then shrunk-fit into the girder of the bridge.

Labeled diagram of a trunnion-hub-girder assembly.

The shrink-fitting is done by first immersing the trunnion in a cold medium such as a dry-ice/alcohol mixture. After the trunnion reaches the steady-state temperature, that is, the temperature of the cold medium, the outer diameter of the trunnion contracts. The trunnion is taken out of the medium and slid through the hole of the hub (Figure \(\PageIndex{2.3}\)).

CAD model showing the trunnion in its contracted state sliding through the hub.

When the trunnion heats up, it expands and creates an interference fit with the hub. In 1995, on one of the bridges in Florida, this assembly procedure did not work as designed. Before the trunnion could be inserted fully into the hub, the trunnion got stuck. Luckily, the trunnion was taken out before it got stuck permanently. Otherwise, a new trunnion and hub would need to be ordered at the cost of \(\$50,000\) . Coupled with construction delays, the total loss could have been more than a hundred thousand dollars.

Why did the trunnion get stuck? Because the trunnion had not contracted enough to slide through the hole. Can you find out why this happened?

Simple Mathematical Model

A hollow trunnion of an outside diameter \(12.363^{\prime\prime}\) is to be fitted in a hub of inner diameter \(12.358^{\prime\prime}\) . The trunnion was put in a dry ice/alcohol mixture (temperature of the fluid - dry-ice/alcohol mixture is \(- 108{^\circ}\text{F}\) ) to contract the trunnion so that it can be slid through the hole of the hub. To slide the trunnion without sticking, a diametrical clearance of at least \(0.01^{\prime\prime}\) is required between the trunnion and the hub. Assuming the room temperature is \(80{^\circ}\text{F}\) , is immersing the trunnion in a dry-ice/alcohol mixture a correct decision?

To calculate the contraction in the diameter of the trunnion, the thermal expansion coefficient at room temperature is used. In that case, the reduction \(\Delta D\) in the outer diameter of the trunnion is

\[\displaystyle\Delta D = D\alpha\Delta T \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.1}) \nonumber\]

\[D = \text{ outer diameter of the trunnion,} \nonumber\]

\[\alpha = \text{ coefficient of thermal expansion coefficient at room temperature, and} \nonumber\]

\[\Delta T = \text{change in temperature.} \nonumber\]

Solution to Simple Mathematical Model

\[D = 12.363^{\prime\prime} \nonumber\]

\[\alpha = 6.47 \times 10^{-6}\ \text{in/in/}^{\circ}\text{F} \text{ at } 80{^\circ}\text{F} \nonumber\]

\[\displaystyle \begin{split} \Delta T&= T_{\text{fluid}} - T_{\text{room}}\\ &= - 108 - 80\\ &= - 188{^\circ}\ \text{F}\end{split} \nonumber\]

\[T_{\text{fluid}}= \text{ temperature of dry-ice/alcohol mixture} \nonumber\]

\[T_{\text{room}}= \text{ room temperature} \nonumber\]

the reduction in the outer diameter of the trunnion from Equation \((\PageIndex{2.1})\) hence is given by

\[\begin{split} \Delta D &= (12.363)\left( 6.47 \times 10^{- 6} \right)\left( - 188 \right)\\ &=- 0.01504^{\prime\prime} \end{split} \nonumber\]

So the trunnion is predicted to reduce in diameter by \(0.01504^{\prime\prime}\) . But is this enough reduction in diameter? As per specifications, the trunnion diameter needs to change by

\[\begin{split} \Delta D &= -\text{trunnion outside diameter} + \text{hub inner diameter} - \text{diametric clearance}\\ &= -12.363 +12.358 - 0.01\\ &= - 0.015^{\prime\prime} \end{split} \nonumber\]

So, according to this calculation, immersing the steel trunnion in dry-ice/alcohol mixture gives the desired contraction of greater than \(0.015^{\prime\prime}\) as the predicted contraction is \(0.01504^{\prime\prime}\) . But, when the steel trunnion was put in the hub, it got stuck. Why did this happen? Was our mathematical model adequate for this problem, or did we create a mathematical error?

Accurate Mathematical Model

As shown in Figure \(\PageIndex{2.4}\) and Table 1, the thermal expansion coefficient of steel decreases with temperature and is not constant over the range of temperature the trunnion goes through. Hence, Equation \((\PageIndex{2.1})\) would overestimate the thermal contraction.

Graph of linear thermal expansion coefficient vs temperature. Thermal expansion increases nonlinearly with increasing temperature.

he contraction in the diameter of the trunnion for which the thermal expansion coefficient varies as a function of temperature is given by

\[ \Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}} \alpha dT \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.2}) \nonumber\]

Solution to More Accurate Mathematical Model

So, one needs to curve fit the data to find the coefficient of thermal expansion as a function of temperature. This curve is found by regression where we best fit a function to the data given in Table 1. In this case, we may fit a second-order polynomial

\[\displaystyle\alpha = a_{0} + a_{1} T + a_{2} T^{2}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.3}) \nonumber\]

The values of the coefficients in the above Equation \((\PageIndex{2.3})\) will be found by polynomial regression (we will learn how to do this later in the chapter on Nonlinear Regression). At this point, we are just going to give you these values, and they are

\[\begin{bmatrix} a_{0} \\ a_{1} \\ a_{2} \\ \end{bmatrix} = \begin{bmatrix} 6.0150 \times 10^{- 6} \\ 6.1946 \times 10^{- 9} \\ - 1.2278 \times 10^{- 11} \\ \end{bmatrix} \nonumber\]

to give the polynomial regression model (Figure \(\PageIndex{2.5}\)) as

\[\displaystyle \begin{split} \alpha &= a_{0} + a_{1}T + a_{2}T^{2}\\ &= {6.0150} \times {1}{0}^{- 6} + {6.1946} \times {10}^{- 9}T - {1.2278} \times {10}^{- {11}}T^{2} \end{split} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.4}) \nonumber\]

Knowing the values of \(a_{0}\) , \(a_{1}\) , and \(a_{2}\) , we can then find the contraction in the trunnion diameter from Equations \((\PageIndex{2.2})\) and \((\PageIndex{2.3})\) as

\[\begin{split} \displaystyle\Delta D &= D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{(a_{0} + a_{1}T + a_{2}T^{2}}){dT}\\ &= D\left\lbrack a_{0}T + a_{1}\frac{T^{2}}{2} + a_{2}\frac{T^{3}}{3} \right\rbrack\begin{matrix} T_{\text{fluid}} \\ \\ T_{\text{room}} \\ \end{matrix}\\ &= D\lbrack a_{0}(T_{\text{fluid}} - T_{\text{room}}) + a_{1}\frac{({T_{\text{fluid}}}^{2} - {T_{\text{room}}}^{2})}{2}\\ & \ \ \ \ \ + a_{2}\frac{({T_{\text{fluid}}}^{3} - {T_{\text{room}}}^{3})}{3}\rbrack\;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{2.5}) \end{split} \nonumber\]

Substituting the values of the variables gives

\[\displaystyle \begin{split} \Delta D &= 12.363\begin{bmatrix} 6.0150 \times 10^{- 6} \times ( - 108 - 80) \\ + 6.1946 \times 10^{- 9}\displaystyle \frac{\left( ( - 108)^{2} - (80)^{2} \right)}{2} \\ - 1.2278 \times 10^{- 11}\displaystyle \frac{(( - 108)^{3} - (80)^{3})}{3} \\ \end{bmatrix}\\ &= - 0.013689^{\prime\prime}\end{split} \nonumber\]

Second-order polynomial regression model for the coefficient of thermal expansion as a function of temperature.

What do we find here? The contraction in the trunnion is not enough to meet the required specification of \(0.015^{\prime\prime}\) .

Implementing the Solution

Although we were able to find out why the trunnion got stuck in the hub, we still need to find and implement a solution. What if the trunnion were immersed in a medium that was cooler than the dry-ice/alcohol mixture of \(- 108{^\circ}F\) , say liquid nitrogen, which has a boiling temperature of \(- 321{^\circ}F\) ? Will that be enough for the specified contraction in the trunnion?

As given in Equation \((\PageIndex{2.5})\)

\[\displaystyle \begin{split} \Delta D &= D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{(a_{0} + a_{1}T + a_{2}T^{2}}){dT}\\ &= D\left\lbrack a_{0}T + a_{1}\frac{T^{2}}{2} + a_{2}\frac{T^{3}}{3} \right\rbrack\begin{matrix} T_{\text{fluid}} \\ \\ T_{\text{room}} \\ \end{matrix}\\ &= D\lbrack a_{0}(T_{\text{fluid}} - T_{\text{room}}) + a_{1}\frac{({T_{\text{fluid}}}^{2} - {T_{\text{room}}}^{2})}{2}\\ & \ \ \ \ \ + a_{2}\frac{({T_{\text{fluid}}}^{3} - {T_{\text{room}}}^{3})}{3}\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2.5}-repeated) \end{split} \nonumber\]

which gives

\[\displaystyle \begin{split} \Delta D &= 12.363\begin{bmatrix} 6.0150 \times 10^{- 6} \times ( - 321 - 80) \\ + 6.1946 \times 10^{- 9} \displaystyle\frac{\left( ( - 321)^{2} - (80)^{2} \right)}{2} \\ \ - 1.2278 \times 10^{- 11} \displaystyle\frac{(( - 321)^{3} - (80)^{3})}{3} \\ \end{bmatrix}\\ & \\ &= - 0.024420^{\prime\prime} \end{split} \nonumber\]

The magnitude of this contraction is larger than the specified value of \(0.015^{\prime\prime}\).

So here are some questions that you may want to ask yourself later in the course.

1) What if the trunnion were immersed in liquid nitrogen (boiling temperature \(= - 321{^\circ}\text{F}\) )? Will that cause enough contraction in the trunnion?

2) Rather than regressing the thermal expansion coefficient data to a second-order polynomial so that one can find the contraction in the trunnion OD, how would you use the trapezoidal rule of integration for unequal segments?

3) What is the relative difference between the two results?

4) We chose a second-order polynomial for regression. Would a different order polynomial be a better choice for regression? Is there an optimum order of polynomial we could use?

Title: Steps of Solving Engineering Problems

Summary : This video teaches you the steps of solving an engineering problem- define the problem, model the problem, solve, and implementation of the solution.

Lesson 3: Overview of Mathematical Processes Covered in This Course

1) enumerate the seven mathematical processes for which numerical methods are used.

Numerical methods are techniques to approximate mathematical processes. This introductory numerical methods course will develop and apply numerical techniques for the following mathematical processes:

1) Roots of Nonlinear Equations

2) Simultaneous Linear Equations

3) Curve Fitting via Interpolation

4) Differentiation

5) Curve Fitting via Regression

6) Numerical Integration

7) Ordinary Differential Equations.

Some undergraduate courses in numerical methods may include topics of partial differential equations, optimization, and fast Fourier transforms as well.

Roots of a Nonlinear Equation

The ubiquitous formula

\[ x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.1}) \nonumber\]

of finding the roots of a quadratic equation \(\displaystyle ax^{2} + bx + c = 0\) goes back to the ancient world. But in the real world, we get equations that are not just the quadratic ones. They can be polynomial equations of a higher order and transcendental equations.

Take an example of a floating ball shown in Figure \(\PageIndex{3.1}\), where you are asked to find the depth to which the ball will get submerged when floating in the water.

A ball of radius R is floating and partially submerged in water to a distance of x.

Assume that the ball has a density of \(600\ \text{kg}/\text{m}^{3}\) and has a radius of \(0.055\ \text{m}\) . On applying the Newtons laws of motion and hence equating the weight of the ball to the buoyancy force, one finds that the depth, \(x\) in meters, to which the ball is underwater and is given by

\[\displaystyle 3.993 \times 10^{- 4} - 0.165x^{2} + x^{3} = 0 \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{2}) \nonumber\]

Equation \((\PageIndex{3.2})\) is a cubic equation that you will need to solve. The equation will have three roots, and the root that is between \(0\ \text{m}\) (just touching the water surface) and \(0.11\ \text{m}\) (almost submerged) would be the depth to which the ball is submerged. The two other roots would be physically unacceptable. Note that a cubic equation with real coefficients can have a set of one real root and two complex roots or a set of three real roots. You may wonder, why could such an application be important? Let’s suppose you are filling in this tank with water, and you are using this ball as a control so that when the ball goes all the way to the top that the flow of the water stops – say in a fish tank that needs replenishing while the owner is away for a few weeks. So, we do need to figure out how much of the ball is submerged underwater.

A cubic equation can be solved exactly by radicals, but it is a tedious process. The same is true but even more complicated for a general fourth-order polynomial equation as well. However, there is no closed-form solution available for a general polynomial equation of fifth-order or more. So, one has to resort to numerical techniques to solve polynomial and other transcendental nonlinear equations (e.g., finding the nonzero roots of \(\tan x = x\) ).

Simultaneous Linear Equations

Ever since you were exposed to algebra, you have been solving simultaneous linear equations.

A rocket going upwards at launch.

Take this problem statement as an example. Suppose the upward velocity of a rocket (Figure \(\PageIndex{3.2}\)) is given at three different times (Table \(\PageIndex{3.1}\)).

The velocity data is approximated by a polynomial as

\[\displaystyle v\left( t \right) = at^{2} + {bt} + c\ {, 5} \leq t \leq {12}.\;\;\;\;\;\;\;\;\;\;\;\;(\PageIndex{3.3}) \nonumber\]

To estimate the velocity at a time that is not given to us, we can set up the equations to find the coefficients \(a,b,c\) of the velocity profile.

The polynomial in Equation (3) is going through three data points \(\left( t_{1},v_{1} \right),\left( t_{2},v_{2} \right),\) and \(\left( t_{3},v_{3} \right)\) where from Table 1.1.3.1

\[\begin{split} t_{1} &= 5,v_{1} = 106.8\\ t_{2} &= 8,v_{2} = 177.2\\ t_{3} &= 12,v_{3} = 600.0 \end{split} \nonumber\]

Requiring that \(v\left( t \right) = at^{2} + bt + c\) passes through the three data points, gives

\[\begin{split} v\left( t_{1} \right) &= v_{1} = at_{1}^{2} + bt_{1} + c\\ v\left( t_{2} \right) &= v_{2} = at_{2}^{2} + bt_{2} + c\\ v\left( t_{3} \right) &= v_{3} = at_{3}^{2} + bt_{3} + c \end{split} \nonumber\]

Substituting the data \(\left( t_{1},\ v_{1} \right),\ \left( t_{2},\ v_{2} \right),\) and \(\left( t_{3},\ v_{3} \right)\) gives

\[\begin{split} a\left( 5^{2} \right) + b\left( 5 \right) + c = 106.8 \\ a\left( 8^{2} \right) + b\left( 8 \right) + c = 177.2 \\ a\left( 12^{2} \right) + b\left( 12 \right) + c = 600.0 \end{split} \nonumber\]

\[\begin{split} 25a + 5b + c = 106.8 \\ 64a + 8b + c = 177.2 \\ 144a + 12b + c = 600.0 \end{split} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.4}) \nonumber\]

Solving a few simultaneous linear equations such as the above set can be done without the knowledge of numerical techniques. However, imagine that instead of three given points, you were given 10 data points. Now the setting up as well as solving the set of 10 simultaneous linear equations without numerical techniques becomes laborious, if not impossible.

Curve Fitting by Interpolation

Interpolation involves that given a function as a set of data points. How does one find the value of the function at points that are not given?. For this, we choose a function, called an interpolant, and make it pass through all the points involved.

You may think that you have already used interpolation in courses such as Thermodynamics and Statistics. After all, it was just taking two points from a table at the back of the textbook or online and finding the value of the function at a point in between by using a straight line.

Take this problem statement as an example. Let’s suppose the upward velocity of a rocket is given at three different times (Table 1.1.3.1).

If one asked you to estimate the velocity at \(7\ \text{s}\) , one might simply use the straight-line formula you are most accustomed to as given below.

Given ( \(t_{1},\ v_{1}\) ) and ( \(t_{2},\ v_{2}\) ), the value of the function \(v\) at \(t\) is given by

\[\displaystyle v = v_{1} + \frac{v_{2} - v_{1}}{t_{2} - t_{1}}(t - t_{1}) \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.5}) \nonumber\]

Although this is possibly enough for courses such as Thermodynamics and Statistics, there are two questions to ask. Is the value calculated accurately, and how accurate is it? To know that, one needs to calculate at least more than one value. In the above example of a rocket velocity vs. time, one can instead use a second-order polynomial interpolant and set up the three equations and three unknowns to find the unknown coefficients, \(a\) , \(b\) , and \(c\) as given in the previous section.

\[v\left( t \right) = at^{2} + {bt} + c{ ,\ 5} \leq t \leq {12}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.6}) \nonumber\]

Then, the resulting second-order polynomial can be used to find the velocity at \(t = 7\ \text{s}\) .

The value obtained from the second-order polynomial (Equation \((\PageIndex{3.6})\)) can be considered to be a new measure of the value of \(v( 7)\) , and the first-order polynomial (Equation \((\PageIndex{3.5})\)) result can be used to determine the accuracy of the results.

Second-order interpolant for velocity vs. time values given in Table 1.1.3.1.

Numerical Differentiation

You have taken a semester-long course in Differential Calculus, where you found derivatives of continuous functions. So let’s suppose somebody gives you the velocity of a rocket as a continuous and at least once differentiable function of time and wants you to find acceleration. Indeed for this particular problem, you can use your differential calculus knowledge to differentiate the velocity function to get the acceleration and put in the value of time, \(t = 7\ \text{s}\) . What if the velocity vs. time is not given as a continuous and at least once differentiable function? Instead, let’s say the function is given at discrete data points (Table 1.1.3.1). How are you then going to find out what the acceleration at \(t = 7\ \text{s}\) ? Do we draw a straight line from \((5,106.8)\) to \((8,177.2)\) and use the straight-line slope as the estimate of acceleration? How do we know that this is adequate? We could incorporate all three points and find a second-order polynomial as given by Equation \((\PageIndex{3.6})\). This polynomial can now be differentiated to estimate the acceleration at \(t = 7\ \text{s}\) . Now the two values can be used to evaluate the accuracy of the calculated acceleration.

Curve Fitting by Regression

When we talked about curve fitting by interpolation, the chosen interpolant needs to go through all the points considered. What happens when we are given many data points, and we instead want a simplified formula to explain the relationship between two variables. See, for example, in Figure \(\PageIndex{3.4a}\), we are given the coefficient of linear thermal expansion data for cast steel as a function of temperature. Looking at the data, one may proclaim that a straight line could explain the data, and that is drawn in Figure \(\PageIndex{3.4b}\). How we draw this straight line is what is called regression. It would be based on minimizing some form of the residuals between what is observed (given data points) and what is predicted (straight line). It does not mean that every time you have data given to you, you draw a straight line. It is possible that a second-order polynomial or a transcendental function other than the first-order polynomial will be a better representation of this particular data. So these are the questions that we will answer when we discuss regression. We will also discuss the adequacy of linear regression models.

Data points for coefficient of linear thermal expansion for cast steel as a function of temperature.

Numerical Integration

You have taken a whole course on integral calculus. Now, why would we need to make numerical approximations of integrals? Just like the standard normal cumulative distribution function

\[\displaystyle\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{x}e^{- t^{2}/2}{dt} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.7}) \nonumber\]

cannot be solved exactly, or when the integrand values are given at discrete data points, we need to use numerical methods of integration.

Trunnion of a fulcrum assembly of a bascule bridge.

In the previous lesson, we looked at the example of contracting the diameter of a trunnion for a bascule bridge fulcrum assembly by dipping it in a mixture of dry ice and alcohol. The contraction is given by

\[\Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{\alpha\ dT} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.8}) \nonumber\]

\[D = \text{outer diameter of the trunnion,} \nonumber\]

\[\alpha = \text{coefficient of linear thermal expansion that is varying with temperature} \nonumber\]

\[T_{\text{room}}= \text{room temperature} \nonumber\]

\[T_{\text{fluid}}= \text{temperature of dry-ice alcohol mixture.} \nonumber\]

Graph of the varying thermal expansion coefficient as a function of temperature for cast steel.

From Figure \(\PageIndex{3.4a}\), one can note that the coefficient of thermal expansion is only given at discrete temperatures and not as a known continuous function that could be integrated exactly. So we have to resort to numerical methods by approximating the data, say, by a second-order polynomial obtained via regression.

In Figure \(\PageIndex{3.6}\), the thermal expansion coefficient of typical cast steel is approximated by a second-order regression polynomial as given by Equation \((\PageIndex{3.9}\)) (how we get this is a later lesson in regression) as

\[\displaystyle\alpha = - 1.2278 \times 10^{- 11}T^{2} + 6.1946 \times 10^{- 9}T + 6.0150 \times 10^{- 6} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.9}) \nonumber\]

The contraction of the diameter then is given by

\[\displaystyle\Delta D = D\int_{T_{\text{room}}}^{T_{\text{fluid}}}{\left( - 1.2278 \times 10^{- 11}T^{2} + 6.1946 \times 10^{- 9}T + 6.015 \times 10^{- 6} \right){dT}} \;\;\;\;\;\;\;\; (\PageIndex{3.10}) \nonumber\]

and can now be calculated using integral calculus.

Numerical Solution of Ordinary Differential Equations

Taking the same example of the trunnion being dipped in a dry-ice/alcohol mixture, one could ask the question - What would the temperature of the trunnion be after dipping it in the mixture for 30 minutes? The model is given by an ordinary differential equation for the temperature \(\theta\) as a function of time, \({t.}\)

\[\displaystyle -{hA} \left(\theta - \theta_{a} \right) = {mC} \frac{d \theta}{dt}\;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.11}) \nonumber\]

\[h= \text{the convective cooling coefficient,}\ \text{W/m}^{2} \cdot \text{K} \nonumber\]

\[A = \text{surface area},\ \text{m}^2 \nonumber\]

\[\theta_{a} = \text{ambient temperature of dry-ice/alcohol mixture},\ \text{K} \nonumber\]

\[m = \text{mass of the trunnion, kg} \nonumber\]

\[C = \text{specific heat of the trunnion,}\ \text{J/(kg} \cdot \text{K)} \nonumber\]

The differential Equation \((\PageIndex{3.11})\) can be solved exactly by using the classical solution, Laplace transform, or separation of variables techniques. So, where do numerical methods enter into the picture for this problem? For the temperature range of room temperatures to cold media such as dry-ice/alcohol, several of the variables in Equation \((\PageIndex{3.11})\) are not constant but change with the temperature. These include the convection coefficient \(h\) as well as the specific heat \(C\) . Now, this differential equation has turned nonlinear as follows.

\[\displaystyle -h(\theta)A\left( \theta - \theta_{a} \right) = mC(\theta)\frac{d \theta}{dt} \;\;\;\;\;\;\;\;\;\;\;\; (\PageIndex{3.12}) \nonumber\]

The ordinary differential Equation \((\PageIndex{3.12})\) cannot be solved by exact methods and would need to be solved by a numerical method.

In the above discussion, we have illustrated the need for numerical methods for each of the seven mathematical processes in the course. In the lessons to follow, we will be developing various numerical techniques to approximate the mathematical processes to calculate acceptable accurate values while calculating associated errors.

Title: Overview of Mathematical Processes Covered in This Course

Summary : This lecture shows you four mathematical procedures that need numerical methods - namely, nonlinear equations, differentiation, simultaneous linear equations, and interpolation.

Multiple Choice Test

(1). Solving an engineering problem requires four steps. In order of sequence, the four steps are

(A) formulate, model, solve, implement

(B) formulate, solve, model, implement

(C) formulate, model, implement, solve

(D) model, formulate, implement, solve

(2). One of the roots of the equation \(x^{3} - 3x^{2} + x - 3 = 0\) is

(C) \(\sqrt{3}\)

(3). The solution to the set of equations

\[\begin{split} 25a + b + c &= 25 \\ 64a + 8b + c &= 71\\ 144a + 12b + c &= 155 \end{split} \nonumber\]

most nearly is \(\left( a,b,c \right) =\)

(A) \((1,1,1)\)

(B) \((1,-1,1)\)

(C) \((1,1,-1)\)

(D) does not have a unique solution.

(4). The exact integral of \(\displaystyle \int_{0}^{\frac{\pi}{4}} 2 \cos 2x \ dx\) is most nearly

(A) \(-1.000\)

(B) \(1.000\)

(C) \(0.000\)

(D) \(2.000\)

(5). The value of \(\displaystyle \frac{dy}{dx}\left( 1.0 \right)\) , given \(y = 2\sin\left( 3x \right)\), is most nearly

(A) \(-5.9399\)

(B) \(-1.980\)

(C) \(0.31402\)

(D) \(5.9918\)

(6). The form of the exact solution of the ordinary differential equation \(\displaystyle 2\frac{dy}{dx} + 3y = 5e^{- x},\ y\left( 0 \right) = 5\) is

(A) \(Ae^{- 1.5x} + Be^{x}\)

(B) \(Ae^{- 1.5x} + Be^{- x}\)

(C) \(Ae^{1.5x} + Be^{- x}\)

(D) \(Ae^{- 1.5x} + Bxe^{- x}\)

For complete solution, go to

http://nm.mathforcollege.com/mcquizzes/01aae/quiz_01aae_introduction_answers.pdf

Mathematics > Numerical Analysis

Title: convergence analysis of a variable projection method for regularized separable nonlinear inverse problems.

Abstract: Variable projection methods prove highly efficient in solving separable nonlinear least squares problems by transforming them into a reduced nonlinear least squares problem, typically solvable via the Gauss-Newton method. When solving large-scale separable nonlinear inverse problems with general-form Tikhonov regularization, the computational demand for computing Jacobians in the Gauss-Newton method becomes very challenging. To mitigate this, iterative methods, specifically LSQR, can be used as inner solvers to compute approximate Jacobians. This article analyzes the impact of these approximate Jacobians within the variable projection method and introduces stopping criteria to ensure convergence. We also present numerical experiments where we apply the proposed method to solve a blind deconvolution problem to illustrate and confirm our theoretical results.

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Numerical Reasoning Tests: Guidelines & Practice Examples

Table of Contents

What is a numerical reasoning test?

A numerical reasoning test is an aptitude test measuring ability to perform calculations and interpret data in the form of charts . There are five common types of numerical reasoning tests: calculation, estimation, number sequence, word problem, and data interpretation . Most of them are in multiple-choice format. To pass the test, candidates have to make quick and accurate calculations as the test is a time constraint.

Calculation, Word problem, and Data Interpretation are the three most common question types in a numerical reasoning test. Number Sequence and Estimation are other types of questions that you may also encounter in a numerical reasoning test, but Number Sequence is often included in inductive reasoning tests and Estimation is more likely to appear in numerical reasoning tests for consulting or banking applicants.

Prevalence of each type: 

Calculation questions in numerical reasoning tests focus on assessing your math operation abilities and therefore are often given in pure calculation forms (no word context). Word problems are short context calculation problems with ENOUGH data and information for delivering the answer ; data interpretation is long context problems with ABUNDANT data and information .

To improve your performance on a numerical reasoning test, you’ll need to equip yourself with the following skills:

  • Make quick and accurate calculations
  • Understand and create formulas in different contexts (daily, business, and finance)
  • Read different types of chart presentations and filter data

As opposed to standardized math tests like GCSE or A-level exams, where you are often required to apply all kinds of mathematical techniques you’ve learned at school in impractical contexts, numerical reasoning tests are more about problem-solving in a realistic setting. These tests simulate the types of analysis you might be expected to perform in your future job.

What does a numerical reasoning test examine?

Numerical reasoning tests are designed to examine the competence of test-takers in solving basic math calculations, analyzing, interpreting, and drawing conclusions from different types of data sets.

Calculation abilities

Calculation skills are measured on numerical reasoning examinations using a variety of question formats. Numerical reasoning exams evaluate your ability to calculate by evaluating how well you do on math problems including decimals, percentages, ratios, etc., as well as fundamental arithmetic operations like addition, subtraction, multiplication, and division.

Simple math questions often don’t test your ability to think or solve problems because such questions only require quick and precise calculations.

In other types of problems, such as data interpretation, calculation abilities are not the primary assessing factor, but rather a way of assisting you in arriving at the ultimate solution.

problem solving numerical analysis

Data analyzing competence

Numerical reasoning tests measure your ability to analyze data in questions that involve math issues in a word context or data presentations. You will be provided either adequate or abundant data to work with for the final answer.

In questions measuring data analysis skills, you must interpret and evaluate data within a given context in order to determine what information and calculations are necessary to answer the question.

Data interpretation skill

Numerical reasoning assessments assess your data interpretation ability by asking you to derive particular inferences from your calculations . These kinds of questions are popular in numerical reasoning assessments used in consulting and business.

Questions measuring interpretation abilities frequently include workplace-based materials relevant to the function and level for which the candidate is seeking, allowing recruiters to determine whether the prospect possesses the necessary skills.

Common question types

There are five types of questions that are asked in a numerical reasoning test, which are calculation, word problem, data interpretation, estimation, and number sequence .

Long and short context word problems and data interpretation are the two most common questions in numerical reasoning tests while the other three types are often adopted in gamified numerical reasoning tests.

Data interpretation questions tend to be more popular for testing high-level roles such as those in management, though they might still be questioned about basic numeracy computation and estimation. 

problem solving numerical analysis

Question Types in the Numerical Reasoning Test (Source: Numerical Reasoning Psychometric Success, Paul Newton & Helen Bristoll)

Word problem

Word problem or numerical comprehension is calculation questions with short context such as daily, business and finance context. You are provided with sufficient data instead of abundant data as distractors like data interpretation. Each question is limited to 1 minute and a calculator is allowed.

Most test suppliers provide numerical reasoning questions under this format because of two reasons. One, it is not too straightforward – candidates need to take an extra step of extracting the numbers before the actual computation. And two, math with a context is more fun and meaningful.

Numerical calculation multiple-choice questions often make use of distractors, which are options that are either very similar to the correct answer or can be achieved by making a common error.

Now, practice with these sample word problems for yourself:

1. If Millie starts school at 9:00 am and finishes at 4:15 pm, and has a total of 75 minutes of break a day, how many hours does she have school in 3 days?

2. In an employee survey of 325 employees, the response rate was six out of 10. How many employees did not complete the survey?

3. A restaurant bill is made up of the following:

Starters: $11.75 Main courses: $34.25 Desserts: $14 10% service charge on total bill

How much is the bill?

4. A consultancy’s operating costs to turnover ratio is 3:20 each year. If the company’s turnover is £213,250 in Year 1, £268,460 in Year 2, and £328,915 in Year 3 what are the total operating costs for the three-year period?

5. At an exchange rate of €1.20 to the £, what is the cost (in £) of three boxes of office paper at €6.60 per box?

4. £121,594

5. £16.50

Data interpretation

The second common question type is data interpretation questions . You may be given numerical data in the forms of tables, graphs, and charts and asked to draw inferences based on the information. Applicants for jobs requiring numerical data analysis or decision-making should be prepared to answer this type of question.

Presentation of data in this kind of question can be classified into 5 basic types :

  • Line charts
  • Caselet (data is provided in a paragraph)

Five simple steps are required when attempting to interpret data:

Step 1: Look through the charts and understand what kind of data they present. If there are multiple charts, make sure you understand the relationship between them.

Step 2: Read the question and determine what kind of data you’ll need.

Step 3: Find the table/graph/chart that has the information you need and locate the information.

Step 4: Apply/Manipulate the information to compute the answer.

Step 5 (optional): Translate the numerical answer into a meaningful inference if the question requires you to do so. 

There are typically two levels of difficulty when it comes to data interpretation questions:

  • Level 1: Single chart (simple)

Questions at the intermediate level usually provide ONE table/chart/graph for you to work with and are pretty straightforward, that is, only one calculation is needed to answer each question.

  • Level 2: Compound chart (advanced)

Advanced data interpretation questions often present two or more sources of information (tables/charts/graphs). They might require at least two calculations to come up with the answer, though these computations aren’t necessarily difficult. It might be tricky to locate the data you need because either there is so much information or you might need to combine charts to extract the data.

Test yourself with these two sets of data interpretation questions:

  • Level 1: Single chart

The figures show a population by economic activity and district.

problem solving numerical analysis

1. How many people in district 1 are employed?

2. How many people across both districts are employed?

3. Express in its simplest form the ratio between economically inactive people and people not economically inactive (those unemployed or employed) in districts 1 and 2 combined.

4. What percentage of people economically inactive across both districts are residents in District 2?

Answer: 

  • Level 2: Advanced

problem solving numerical analysis

1. In which two months is there the same difference between the high street and retail sales for Product A?

A. January & February

B. January & August

C. February & August

D. April & May

E. None of these

2. In which month was high street retail sales higher than retail park sales for Product A?

B. February

3. Between which months did both the retail park and high street sales of Product A fall?

A. February to March

B. March to April

C. April to May

D. May to June

E. June to July

4. In 2008 what is the difference between the UK and Asian sales (to the nearest £ million)?

A. £113 million

B. £123 million

C. £133 million

D. £143 million

E. £153 million

5. In which two quarters were the total non-UK sales the same?

A. Quarter 1 & 2

B. Quarter 1 & 3

C. Quarter 1 & 4

D. Quarter 2 & 3

E. Quarter 3 & 4

6. Due to increasing inflation, 2009’s quarterly non-UK sales are predicted to be 3%, 4%, 5%, and 6% higher than the respective 2008 quarters. What is the total sales prediction for 2009?

A. £544.4 million

B. £555.5 million

C. £566.6 million

D. £577.7 million

E. £588.8 million

Skills to practice for numerical reasoning tests

To handle numerical reasoning test questions, it’s necessary that you have the following skills:

Processing math in different contexts

Filtering data in different chart forms , making quick and accurate calculations/estimations.

In your numerical reasoning tests, you will most likely be asked to perform a lot of arithmetic calculations in various contexts . There are a great number of graphs, tables, and word problems depicting various daily, business, or finance-related contexts that you must work on for the subsequent questions. Each set of provided data might include between two and five questions, each with four or five answer choices. Your job is to determine the accurate answer based entirely on the numerical data presented.

If you want to pass your numerical reasoning test, you must learn to process arithmetic in a variety of contexts. In general, there are three primary math settings that you may encounter: daily, business, and financial.

Filtering data in various formats is what allows you to cope with data interpretation questions (long context) when there is a lot of distracting data . The abundance of details presented in the question rather than the calculations required for the final answer choice is what makes data interpretation complex.

Since data interpretation questions are typically presented in various data forms such as tables, bars, lines, pie charts, and caselets, you can practice filtering data by attempting to solve problems in such data formats. We offer a handful of examples to help you understand how you may try to answer these questions.

problem solving numerical analysis

Your ability to do quick and precise calculations determines whether or not you score high on the test. This ability allows you to perform simple calculations faster to save time for answering more complex math problems.

The key to becoming an expert at completing quick and correct calculations is to practice mental math and basic to advanced calculations such as big numbers, decimals, and serial calculations on a regular basis.

Math formulas in numerical reasoning tests

Besides four common math operations, you’ll need a couple more math formulas when dealing with word problems and data interpretation questions. Below is the list of math formulas you need for a numerical reasoning test.

problem solving numerical analysis

Popular numerical reasoning test providers

There are a number of popular numerical reasoning test providers. The list includes the following test providers:

  • Kenexa (IBM)
  • TalentLens (Pearson)
  • Criteria Corp

In numerical reasoning tests of such test providers, word problems and data interpretation are the two most common question types. Here’s the question prevalence breakdown:

problem solving numerical analysis

Scoring in the McKinsey PSG/Digital Assessment

The scoring mechanism in the McKinsey Digital Assessment

IMAGES

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  1. Problem Solving

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