201 Numerical Methods Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Numerical Methods is the branch of mathematics that builds practical computational tools for solving problems that cannot be resolved by exact analytical means. Many engineering problems involve equations, integrals, or differential equations for which closed-form solutions either do not exist or are too complex to be useful. Numerical methods provide systematic algorithms that produce accurate approximate solutions, and the theory behind those algorithms: convergence, accuracy, stability, and error  is what the engineering Mathematics board exam tests.

This topic appears across multiple engineering disciplines in the licensure examinations. Civil engineers use numerical methods to analyze indeterminate structures and solve boundary value problems. Electrical and electronics engineers apply them in circuit simulation, signal processing, and control system analysis. Mechanical engineers rely on them for heat transfer, fluid mechanics, and structural analysis problems that resist analytical treatment. The board exam mathematics section covers the foundational algorithms directly namely bisection method, Newton-Raphson, interpolation formulas, Simpson’s rule, and Runge-Kutta while the major subject examinations embed numerical reasoning in applied contexts.

This glossary covers 201 numerical methods terms drawn from the complete scope as taught in Philippine engineering programs and reviewed for the licensure examination. The entries span the core topic areas: error analysis, root-finding methods, systems of linear equations, interpolation and curve fitting, numerical differentiation, numerical integration, and numerical solutions to ordinary differential equations. Each definition is written to be technically precise and board exam relevant. Terms that carry their own formulas or computational procedures are described with enough specificity that you can reconstruct the method from the definition alone.

Approach this topic with a dual focus. First, understand what each method does and when to use it. Second, know the formulas well enough to apply them by hand to small problems, because that is exactly how they are tested on the board exam. The algorithms covered here are not complex to execute on small examples. The bisection, Newton-Raphson, trapezoidal rule, and Euler’s method can all be carried out in a few steps with a calculator. What trips reviewees up is not the arithmetic but the vocabulary and the selection logic. This glossary is designed to close that gap.

The 201 Numerical Methods Terms and Definitions

1. 3/8 Rule

See Simpson’s 3/8 Rule. The four-point Newton-Cotes formula using three subintervals with the pattern of weights 1, 3, 3, 1.

2. Absolute Error

The absolute value of the difference between the exact value and the approximate value: |exact − approximate|. Absolute error gives the magnitude of the error without regard to direction and is the most basic measure of accuracy. It is used when the scale of the quantity is known and relevant.

3. Acceleration of Convergence

A technique applied to a slowly converging sequence to produce a faster-converging sequence with the same limit. Aitken’s delta-squared process and Richardson extrapolation are common acceleration methods. Accelerating convergence reduces the number of iterations needed to reach a desired accuracy.

4. Adams-Bashforth Method

An explicit multi-step method for solving ordinary differential equations that uses previously computed solution values to predict the next step without solving an implicit equation. The two-step Adams-Bashforth formula is y_{n+1} = yₙ + h(3f(tₙ, yₙ) − f(t_{n−1}, y_{n−1}))/2. It requires a starting procedure to generate the first few values.

5. Adams-Moulton Method

An implicit multi-step method for solving ordinary differential equations that uses the function value at the new step in the update formula, requiring an iterative or predictor-corrector approach. It is more accurate than the corresponding Adams-Bashforth method of the same order and is often paired with it in a predictor-corrector scheme.

6. Adams Predictor-Corrector

The pairing of the Adams-Bashforth explicit formula as the predictor with the Adams-Moulton implicit formula as the corrector. This combination achieves high accuracy efficiently by avoiding the need to solve the implicit equation iteratively — instead, the explicit prediction is substituted directly into the corrector formula once.

7. Adaptive Integration

A numerical integration strategy that automatically adjusts the step size or number of subintervals based on the local behavior of the integrand, concentrating computational effort where the function varies rapidly. Adaptive methods achieve a specified accuracy more efficiently than fixed-step methods.

8. Adaptive Step Size

A strategy in numerical methods where the step size h is automatically adjusted during computation based on estimated local error. Smaller steps are used where the solution changes rapidly and larger steps where it is smooth. Runge-Kutta-Fehlberg is the standard example of an adaptive step size ODE solver.

9. Aitken’s Delta-Squared Process

An extrapolation technique for accelerating the convergence of a linearly converging sequence. Given three successive iterates x₀, x₁, x₂, the accelerated estimate is x̂ = x₀ − (x₁ − x₀)²/(x₂ − 2x₁ + x₀). It is particularly effective for sequences generated by fixed-point iteration.

10. Algorithm

A finite sequence of well-defined instructions for solving a computational problem or class of problems. In numerical methods, an algorithm specifies the exact steps: initialization, iteration, convergence check, and output  needed to compute an approximate solution. The efficiency, accuracy, and stability of a numerical method are properties of its algorithm.

11. Approximate Solution

The output of any numerical method, representing a computed estimate of the exact mathematical solution. All numerical methods produce approximate solutions, and the goal of error analysis is to bound and control the difference between the approximate and exact values.

12. Approximation

A value or function that is close to but not exactly equal to the true value or function. All numerical methods produce approximations rather than exact answers. The quality of an approximation is measured by the error between the approximate and exact values.

13. Back Substitution

The process of solving a system of linear equations after it has been reduced to upper triangular form by Gaussian elimination. Starting from the last equation (which has one unknown), each variable is solved in sequence by substituting previously determined values upward through the system.

14. Backward Difference

A finite difference approximation that estimates derivatives using the current point and points to the left (earlier in the sequence). The first backward difference of f at xₙ is ∇f(xₙ) = f(xₙ) − f(xₙ₋₁). Backward differences are used in backward difference interpolation formulas and implicit ODE methods.

15. Backward Euler Method

An implicit first-order method for solving ordinary differential equations, defined by y_{n+1} = yₙ + h·f(t_{n+1}, y_{n+1}). Because y_{n+1} appears on both sides, the method requires solving an equation at each step, but it is unconditionally stable for stiff problems, making it preferred over the explicit forward Euler method in those cases.

16. Bairstow’s Method

A numerical method for finding all roots of a polynomial, including complex roots, by iteratively extracting quadratic factors using synthetic division. It avoids working with complex arithmetic directly and converges quadratically near the roots. It generalizes Newton’s method to polynomial root extraction.

17. Banded Matrix

A sparse matrix in which all non-zero entries are confined to a diagonal band around the main diagonal. Banded matrices arise naturally in finite difference and finite element methods. Specialized algorithms exploit the banded structure to solve linear systems much more efficiently than general methods.

18. Bisection Method

A root-finding method that repeatedly halves a bracketing interval [a, b] in which the function changes sign, guaranteeing the presence of a root. At each step, the midpoint c = (a + b)/2 is computed and the interval is replaced by whichever half still contains the sign change. It converges linearly and is the most reliable but slowest of the standard root-finding methods.

19. Boundary Condition

A condition that a solution must satisfy at the boundary of the domain. In ordinary differential equations, boundary conditions are specified at two or more distinct points. Boundary value problems are more complex to solve numerically than initial value problems and require methods such as shooting or finite differences.

20. Boundary Value Problem

A differential equation problem in which conditions are specified at two or more boundary points of the domain rather than all at a single initial point. Numerical methods for boundary value problems include the shooting method, finite difference method, and finite element method.

21. Bracketing Method

Any root-finding method that maintains an interval [a, b] known to contain a root, based on the intermediate value theorem (f(a) and f(b) have opposite signs). Bisection and regula falsi are bracketing methods. They are guaranteed to converge but generally slower than open methods like Newton-Raphson.

22. Central Difference

A finite difference approximation that estimates a derivative using values symmetrically placed around the point of interest. The first central difference approximation of f'(x) is [f(x + h) − f(x − h)] / (2h). Central differences are more accurate than forward or backward differences for the same step size, with error of order h².

23. Central Difference Formula for Second Derivative

The finite difference approximation f”(x) ≈ [f(x + h) − 2f(x) + f(x − h)] / h², obtained by applying the central difference operator twice. It has second-order accuracy (error of order h²) and is the standard formula for approximating second derivatives in finite difference methods for differential equations.

24. Chebyshev Nodes

Specific interpolation nodes chosen to minimize the maximum interpolation error over an interval. They are the roots of Chebyshev polynomials: xₖ = cos((2k − 1)π/(2n)) for k = 1, 2, …, n, mapped to the interval [a, b]. Using Chebyshev nodes significantly reduces the Runge phenomenon in polynomial interpolation.

25. Chebyshev Polynomials

A family of orthogonal polynomials defined by Tₙ(x) = cos(n arccos x) on the interval [−1, 1]. They have the minimax property: among all polynomials of degree n with leading coefficient 1, the Chebyshev polynomial has the smallest maximum absolute value. They are used in polynomial approximation and numerical integration.

26. Chord Method

See Secant Method. A root-finding method that approximates the derivative in Newton’s formula using a finite difference between two previous function values.

27. Closed Integration Formula

A numerical integration rule that uses function values at both endpoints of the integration interval, as opposed to open formulas which exclude the endpoints. The trapezoidal rule and Simpson’s rule are closed Newton-Cotes formulas. Closed formulas are generally more accurate than open formulas of the same order.

28. Composite Integration Rule

A numerical integration method applied over the entire interval by dividing it into multiple subintervals and applying a basic rule (trapezoidal, Simpson’s, etc.) to each subinterval. Composite rules achieve higher accuracy by using more subintervals rather than higher-degree polynomials. The composite trapezoidal and composite Simpson’s rules are the most common.

29. Composite Simpson’s Rule

The application of Simpson’s rule over n equal subintervals (n must be even), giving ∫f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(x_{n−1}) + f(xₙ)], where h = (b − a)/n. The composite Simpson’s rule has fourth-order accuracy and is one of the most tested integration formulas on the board exam.

30. Composite Trapezoidal Rule

The application of the trapezoidal rule over n equal subintervals, giving ∫f(x)dx ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(x_{n−1}) + f(xₙ)], where h = (b − a)/n. It has second-order accuracy and systematically underestimates or overestimates depending on the concavity of the integrand.

31. Condition Number

A measure of how sensitive the solution of a linear system Ax = b is to small perturbations in A or b. A large condition number indicates an ill-conditioned system where small errors in the input produce large errors in the solution. It is defined as κ(A) = ‖A‖ · ‖A⁻¹‖ and determines the loss of significant digits in numerical solutions.

32. Convergence

The property of a sequence of iterates produced by a numerical method approaching the exact solution as the number of iterations increases. A method converges if the error decreases toward zero. Convergence analysis determines when and how quickly a method reaches an acceptable solution.

33. Convergence Criterion

A stopping condition used to terminate an iterative numerical method when the solution is sufficiently accurate. Common criteria include |x_{n+1} − xₙ| < tolerance (successive approximation), |f(xₙ)| < tolerance (function value), or a maximum number of iterations. The criterion must be chosen carefully to balance accuracy and computational cost.

34. Convergence of Iterative Methods

The property that the sequence of iterates x₀, x₁, x₂, … produced by an iterative method approaches the true solution x* as the number of iterations increases. Convergence depends on the method, the problem, and the initial guess. The rate of convergence (linear, quadratic, superlinear) determines how quickly the method reaches a desired accuracy.

35. Corrector Formula

In a predictor-corrector method, the implicit formula used to improve (correct) the prediction made by the explicit predictor formula. The corrector uses the predicted value to evaluate the function at the new step and then applies the implicit update. The Adams-Moulton formula is a common corrector.

36. Crout’s Method

A variant of LU decomposition that directly computes the lower triangular matrix L and upper triangular matrix U of a coefficient matrix without partial pivoting. Crout’s method is used to solve linear systems efficiently by decomposing the system once and then performing forward and back substitution for any right-hand side vector.

37. Cubic Spline

A piecewise cubic polynomial that interpolates a set of data points with continuous first and second derivatives at the interior knots. Natural cubic splines additionally set the second derivative to zero at the two endpoints. Cubic splines produce smooth, visually appealing interpolants without the oscillation problems of high-degree polynomials.

38. Curve Fitting

The process of finding a mathematical function (line, polynomial, exponential, etc.) that best represents the overall trend in a set of data points, without requiring exact passage through every point. Curve fitting minimizes a measure of overall error, typically the sum of squared residuals. It differs from interpolation, which requires exact passage through all data points.

39. Data Point

A pair (xᵢ, yᵢ) consisting of an independent variable value and its corresponding observed or computed dependent variable value. Data points are the inputs to interpolation, curve fitting, and numerical integration methods. The accuracy and spacing of data points strongly influence the quality of numerical results.

40. Dead-Load Iteration

An iterative correction scheme commonly used in structural engineering numerical analysis where the solution is updated repeatedly using residual forces from the previous iteration. It is a specific application of fixed-point iteration in the context of nonlinear structural problems.

41. Deflation

The process of reducing a polynomial to a lower-degree polynomial after one root has been found, by dividing out the corresponding linear factor using synthetic division. Deflation allows the remaining roots to be found from the reduced polynomial. Accumulated rounding errors can cause deflation to become less accurate for later roots.

42. Degree of Precision

The highest degree polynomial for which a numerical integration formula gives an exact result. The trapezoidal rule has degree of precision 1 (exact for polynomials up to degree 1). Simpson’s rule has degree of precision 3 (exact for polynomials up to degree 3). Higher degree of precision means greater accuracy for smooth functions.

43. Derivative Approximation

The numerical estimation of a derivative at a point using finite differences. Forward, backward, and central difference formulas are the standard approaches. The accuracy of the approximation depends on the step size h and the order of the formula, with central differences being most accurate for the same h.

44. Deterministic Algorithm

An algorithm that produces the same output every time it is run with the same input. All standard numerical methods: bisection, Gaussian elimination, Runge-Kutta are all deterministic. This contrasts with stochastic or randomized methods like Monte Carlo simulation.

45. Difference Table

A tabular arrangement of forward, backward, or central differences of a function, used in Newton’s interpolation formulas and in deriving numerical differentiation and integration rules. Constructing a difference table is the first step in applying Gregory-Newton interpolation.

46. Differential Equation (Numerical Solution)

The approximate solution of a differential equation obtained by replacing derivatives with finite difference approximations and advancing the solution step by step from known initial or boundary conditions. The accuracy and stability of the numerical solution depend on the method, step size, and the nature of the equation.

47. Differentiation Matrix

A matrix D such that the vector of derivative values at grid points equals D times the vector of function values at those grid points. Differentiation matrices are used in spectral methods and collocation methods for differential equations. Their construction is based on differentiation of the interpolating polynomial through the grid points.

48. Divided Difference

The generalized slope of a function defined recursively over a set of nodes, used in Newton’s divided difference interpolating polynomial. The first divided difference is f[xᵢ, xⱼ] = (f(xⱼ) − f(xᵢ))/(xⱼ − xᵢ). Higher-order divided differences are computed recursively and form the coefficients of Newton’s interpolating polynomial.

49. Dominant Eigenvalue

The eigenvalue of a matrix with the largest absolute value. Power iteration converges to the dominant eigenvalue and its corresponding eigenvector. Knowledge of the dominant eigenvalue is important in stability analysis, principal component analysis, and convergence analysis of iterative methods.

50. Double Integration (Numerical)

The numerical approximation of a double integral ∬f(x, y) dA over a two-dimensional region. It is typically implemented by applying one-dimensional integration rules repeatedly, integrating over one variable at fixed values of the other. The error analysis extends naturally from the one-dimensional case.

51. Eigenvalue Problem (Numerical)

The problem of computing the eigenvalues and eigenvectors of a matrix using iterative numerical methods. Power iteration finds the dominant eigenvalue. The QR algorithm finds all eigenvalues. Numerical eigenvalue computation is important in vibration analysis, stability assessment, and principal component analysis.

52. Elimination Method

A method for solving a system of linear equations by systematically eliminating variables using row operations. Gaussian elimination is the standard elimination method. It reduces the system to upper triangular form from which the solution is obtained by back substitution.

53. Empirical Formula

A mathematical expression derived from experimental data rather than from theoretical derivation. Empirical formulas are developed using curve fitting or regression techniques and are valid within the range of the data used to construct them. They are commonly used in engineering design and analysis.

54. Error

The difference between the exact solution and the numerical approximation. Error in numerical methods arises from two primary sources: truncation error (from approximating mathematical operations) and round-off error (from finite precision arithmetic). Controlling and estimating error is central to the practice of numerical methods.

55. Error Analysis

The study of the sources, magnitudes, propagation, and control of errors in numerical computations. A thorough error analysis classifies the type of error, derives an expression for its magnitude, and determines how the error behaves as step size or number of iterations changes.

56. Error Bound

An upper limit on the magnitude of the error in a numerical approximation, derived from mathematical analysis of the method. For the composite Simpson’s rule, the error bound is |E| ≤ (b − a)⁵ f⁽⁴⁾(ξ) / (180n⁴) for some ξ in [a, b]. Error bounds guide the selection of step size to meet accuracy requirements.

57. Error Propagation

The process by which errors introduced at one stage of a computation influence the errors at later stages. In iterative methods, error propagation determines whether errors amplify (unstable) or diminish (stable). In direct methods, error propagation through arithmetic operations determines the final accuracy.

58. Euler’s Method

The simplest explicit first-order method for solving initial value problems of the form y’ = f(t, y), y(t₀) = y₀. The update formula is y_{n+1} = yₙ + h·f(tₙ, yₙ), where h is the step size. Euler’s method has first-order accuracy (error proportional to h) and is the foundation for understanding more advanced ODE solvers. It is among the most tested numerical methods on the board exam.

59. Explicit Method

A numerical method in which the solution at the next step is computed directly from known values at the current and previous steps, without solving an equation. Euler’s method and Adams-Bashforth are explicit methods. Explicit methods are easy to implement but may be unstable for stiff problems with large step sizes.

60. Extrapolation

The estimation of a function value outside the range of known data, or the use of multiple approximations at different step sizes to construct a more accurate result. Richardson extrapolation is the standard technique for improving accuracy by combining two approximations with different step sizes to cancel leading error terms.

61. False Position Method

See Regula Falsi Method.

62. Finite Difference

An approximation to a derivative obtained by dividing the difference of function values at nearby points by the spacing between those points. Finite differences are the fundamental building blocks of numerical differentiation, numerical integration formulas, and finite difference methods for differential equations.

63. Finite Difference Method

A method for solving differential equations by replacing derivatives with finite difference approximations at a set of grid points, converting the differential equation into a system of algebraic equations. It is widely used for boundary value problems and partial differential equations in engineering analysis.

64. Finite Element Method

A powerful numerical technique for solving boundary value problems by dividing the domain into small subdomains called elements, approximating the solution within each element using simple basis functions, and assembling the resulting system of equations. It is the standard computational tool in structural, thermal, and fluid mechanics engineering analysis.

65. Fixed-Point

A value x* such that g(x*) = x*, i.e., the function maps the point to itself. Root-finding problems f(x) = 0 are converted to fixed-point problems by writing x = g(x) = x − f(x)/k for some constant k. Fixed-point iteration converges to x* when |g'(x*)| < 1.

66. Fixed-Point Iteration

A root-finding method that rewrites the equation f(x) = 0 in the form x = g(x) and iterates x_{n+1} = g(xₙ) starting from an initial guess x₀. The method converges when |g'(x*)| < 1 near the root x*. The convergence rate is linear (first order) when |g'(x*)| is close to 1 and approaches superlinear when |g'(x*)| is close to 0.

67. Fixed-Point Theorem

The theorem guaranteeing that if g is a contraction mapping on a closed interval (meaning |g'(x)| ≤ k < 1 for all x in the interval and g maps the interval to itself), then the fixed-point iteration x_{n+1} = g(xₙ) converges to a unique fixed point. It provides the theoretical foundation for the convergence of fixed-point iteration.

68. Forward Difference

A finite difference approximation using the current point and the next point: Δf(xₙ) = f(x_{n+1}) − f(xₙ). The forward difference approximation to f'(x) is [f(x + h) − f(x)]/h with first-order error O(h). Forward differences are the basis of Newton’s forward interpolation formula and the explicit Euler method.

69. Forward Elimination

The first phase of Gaussian elimination in which elementary row operations are applied to the augmented matrix to create zeros below each pivot, transforming the system to upper triangular form. Forward elimination proceeds column by column from left to right.

70. Forward Euler Method

See Euler’s Method. The explicit first-order ODE solver y_{n+1} = yₙ + h·f(tₙ, yₙ).

71. Fourth-Order Runge-Kutta Method

The most widely used single-step method for solving initial value problems, combining four function evaluations per step to achieve fourth-order accuracy (error proportional to h⁴). The classical formula uses weighted averages of slopes k₁, k₂, k₃, k₄ evaluated at the beginning, two midpoints, and end of the step. It is the standard ODE solver in most engineering applications and is heavily tested on the board exam.

72. Frobenius Norm

A matrix norm computed as the square root of the sum of squares of all matrix entries. It is used in error analysis and in measuring the size of matrices in numerical computations. Unlike operator norms, the Frobenius norm is computed entry-wise.

73. Full Pivoting

A variant of Gaussian elimination in which both rows and columns are searched to find the largest available pivot at each step. Full pivoting provides maximum numerical stability but requires tracking both row and column permutations. Partial pivoting (row pivoting only) is the more common compromise between stability and simplicity.

74. Gauss-Jordan Elimination

An extension of Gaussian elimination that eliminates entries both below and above each pivot, reducing the augmented matrix all the way to reduced row echelon form. The solution is read off directly without back substitution. Gauss-Jordan is used both for solving linear systems and for computing matrix inverses.

75. Gauss-Legendre Quadrature

A Gaussian quadrature rule using Legendre polynomial roots as nodes and specifically derived weights to achieve the highest possible degree of precision for a given number of function evaluations. An n-point Gauss-Legendre rule integrates polynomials of degree up to 2n − 1 exactly.

76. Gauss-Seidel Method

An iterative method for solving a system of linear equations in which each variable is updated using the most recently computed values of all other variables. It generally converges faster than the Jacobi method and is guaranteed to converge for diagonally dominant or positive definite systems. Each new value is used immediately rather than waiting for the entire iteration to complete.

77. Gaussian Elimination

The standard direct method for solving systems of linear equations by applying elementary row operations to the augmented matrix to reduce it to upper triangular form, followed by back substitution. It is O(n³) in computational complexity and is the foundation of most direct linear system solvers.

78. Gaussian Integration Points

The specific nodes (abscissae) used in Gaussian quadrature, chosen as the roots of orthogonal polynomials (Legendre, Laguerre, Hermite, etc.) appropriate to the integration domain and weight function. The optimal placement of these nodes is what gives Gaussian quadrature its superior accuracy compared to Newton-Cotes rules.

79. Gaussian Quadrature

A class of numerical integration methods that select both the nodes (evaluation points) and weights optimally to maximize accuracy for a given number of function evaluations. Gaussian quadrature achieves degree of precision 2n − 1 with n function evaluations, compared to degree n − 1 for Newton-Cotes rules with n points.

80. Global Error

The total accumulated error in a numerical ODE solution at a given point, resulting from the accumulation of local truncation errors over all previous steps. Global error analysis determines how errors propagate and accumulate from the initial condition to the final solution point.

81. Global Truncation Error

The overall error in a numerical method accumulated over all steps from the initial point to the current point. For Euler’s method, the global truncation error is O(h)  first order. For the fourth-order Runge-Kutta method, it is O(h⁴). Global error is one order higher in h than local truncation error.

82. Gregory-Newton Interpolation

See Newton’s Forward Difference Interpolation Formula.

83. Grid Point

A specific location in the discrete set of values at which a numerical method evaluates or stores the solution. In ODE solvers, grid points are the time steps tₙ = t₀ + nh. In finite difference methods for PDEs, grid points form a mesh covering the spatial domain.

84. Heun’s Method

A second-order predictor-corrector ODE method that first predicts y{n+1} = yₙ + h·f(tₙ, yₙ) (Euler step) and then corrects using the average slope: y{n+1} = yₙ + (h/2)[f(tₙ, yₙ) + f(t_{n+1}, y_{n+1})]. Also called the improved Euler method or modified Euler method. It has second-order accuracy (error proportional to h²).

85. Higher-Order Method

A numerical method with a higher-order truncation error, meaning the error decreases more rapidly as step size h decreases. A method of order p has error proportional to h^p. Higher-order methods require fewer steps to achieve a given accuracy but may require more function evaluations per step.

86. Horner’s Method

A computationally efficient algorithm for evaluating a polynomial p(x) = aₙxⁿ + … + a₁x + a₀ by rewriting it in nested form: p(x) = (…((aₙx + a_{n−1})x + a_{n−2})x + … + a₁)x + a₀. It reduces the number of multiplications from O(n²) to O(n) and is the basis of synthetic division.

87. Ill-Conditioned System

A system of linear equations for which small perturbations in the coefficient matrix or right-hand side produce large changes in the solution. An ill-conditioned system has a large condition number. Numerical solutions to ill-conditioned systems suffer from severe loss of significant digits and may be unreliable.

88. Implicit Method

A numerical method in which the solution at the next step appears on both sides of the update equation, requiring an additional solve step at each iteration. Implicit methods are typically more stable than explicit methods for stiff problems. The backward Euler method and Crank-Nicolson method are implicit ODE solvers.

89. Incremental Search Method

A preliminary step in root finding in which the domain is scanned at regular intervals to detect sign changes in f(x), locating subintervals that each contain a root. Once brackets are identified, a more precise method such as bisection or Newton-Raphson is applied within each bracket.

90. Initial Condition

The value of the unknown function (and possibly its derivatives) at the initial point of an initial value problem. For y’ = f(t, y), the initial condition y(t₀) = y₀ specifies where the solution starts. All explicit single-step ODE methods advance the solution step by step starting from the initial condition.

91. Initial Value Problem

A differential equation together with specified values of the unknown function (and derivatives) at a single initial point. IVPs are solved by marching forward from the initial condition using step-by-step methods such as Euler, Heun, or Runge-Kutta.

92. Instability of Numerical Differentiation

The tendency for numerical differentiation to amplify round-off errors as the step size h decreases. While truncation error decreases with h, round-off error increases as h becomes very small because subtraction of nearly equal numbers loses significant digits. An optimal h balances these two competing sources of error.

93. Integration by Parts (Numerical)

A technique used to handle integrals of products of functions in the context of numerical integration. It is analogous to the analytical integration by parts formula and is sometimes used to transform difficult integrals into forms more amenable to numerical treatment.

94. Interpolating Polynomial

A polynomial that passes exactly through a given set of data points (xᵢ, yᵢ). By the Weierstrass theorem and uniqueness of polynomial interpolation, there is exactly one polynomial of degree at most n − 1 passing through n data points. Newton’s and Lagrange’s formulas provide two ways to construct this unique polynomial.

95. Interpolation

The process of estimating the value of a function at a point within the range of known data points, using the assumption that the function can be approximated by a polynomial or other smooth function through those points. Interpolation is distinct from extrapolation (estimation outside the data range) and from regression (fitting without exact passage through points).

96. Interval Halving

See Bisection Method. The repeated halving of an interval to locate a root of a function.

97. Inverse Interpolation

The process of finding the value of the independent variable x corresponding to a specified value of the dependent variable y, using an interpolation formula. It is the reverse of ordinary interpolation and may be solved directly or by applying an interpolation formula to the inverse function.

98. Inverse Power Method

An iterative method for finding the eigenvalue of a matrix with the smallest absolute value and its corresponding eigenvector. It applies the power method to the inverse of the matrix, so the dominant eigenvalue of A⁻¹ corresponds to the smallest eigenvalue of A.

99. Iterative Method

A computational procedure that starts from an initial approximation and generates a sequence of improved approximations by repeatedly applying a fixed formula. Iterative methods are used for root finding (Newton-Raphson, bisection), solving linear systems (Jacobi, Gauss-Seidel), and solving ODEs (Picard iteration).

100. Iterative Refinement

A technique for improving the accuracy of a solution to a linear system obtained by Gaussian elimination, by computing the residual, solving for a correction, and adding the correction to the current solution. It is effective for improving solutions that have been degraded by round-off errors in ill-conditioned systems.

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101. Jacobi Method

An iterative method for solving a system of linear equations in which all variables are updated simultaneously using values from the previous iteration only. It is simpler than the Gauss-Seidel method but generally converges more slowly. It converges for strictly diagonally dominant systems.

102. Jacobian Matrix

The matrix of all first-order partial derivatives of a vector-valued function with respect to its input variables. In Newton’s method for nonlinear systems, the Jacobian plays the role that the derivative plays in the scalar case. Its determinant (the Jacobian determinant) appears in change-of-variable formulas for multiple integrals.

103. Kutta’s Method

An early formulation of the Runge-Kutta idea by Wilhelm Kutta, contributing to the development of the classical fourth-order Runge-Kutta method. The term is sometimes used to refer to the third-order Runge-Kutta method specifically.

104. Lagrange Interpolating Polynomial

A form of the interpolating polynomial expressed as a linear combination of Lagrange basis polynomials: P(x) = Σ yᵢ Lᵢ(x), where Lᵢ(x) = Πⱼ≠ᵢ (x − xⱼ)/(xᵢ − xⱼ). Each basis polynomial Lᵢ(x) equals 1 at xᵢ and 0 at all other nodes. It is conceptually clean and easy to apply for small datasets on board exam problems.

105. Least Squares Fitting

A curve fitting method that determines the parameters of a model by minimizing the sum of the squared residuals between the observed data and the model predictions. It is the standard approach for fitting linear, polynomial, and other models to data. For linear models, the normal equations provide the exact solution.

106. Linear Convergence

A rate of convergence in which the error at each step is a constant fraction of the error at the previous step: |eₙ₊₁| ≈ C|eₙ| for some constant C < 1. The bisection method and fixed-point iteration (when |g'(x*)| > 0) converge linearly. Linear convergence is slower than quadratic convergence.

107. Linear Interpolation

Interpolation using a first-degree polynomial (a straight line) connecting two adjacent data points. The formula is f(x) ≈ f(x₀) + [(f(x₁) − f(x₀))/(x₁ − x₀)](x − x₀). It is the simplest interpolation method and is exact only for linear functions. It underlies the trapezoidal rule for numerical integration.

108. Linear System

A system of equations in which all unknown variables appear to the first power and no products of unknowns appear. Systems of the form Ax = b, where A is a matrix of coefficients, x is the vector of unknowns, and b is a vector of constants, are linear systems. Direct and iterative numerical methods both apply to linear systems.

109. Local Truncation Error

The error introduced in a single step of a numerical ODE method, measuring how well the numerical update formula agrees with the exact Taylor series expansion of the solution. A method is of order p if the local truncation error is O(h^(p+1)). Local truncation error is the per-step accuracy measure.

110. LU Decomposition

The factorization of a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U: A = LU. It is a matrix form of Gaussian elimination. Once the decomposition is computed, systems with different right-hand side vectors can be solved efficiently using forward and back substitution.

111. Machine Epsilon

The smallest positive floating-point number ε such that the computed value of 1 + ε is distinguishable from 1 in the floating-point arithmetic of a given computer. It defines the precision of the floating-point system and determines the minimum achievable round-off error. For IEEE double precision, machine epsilon is approximately 2.2 × 10⁻¹⁶.

112. Matrix Norm

A scalar measure of the size of a matrix, used in error analysis and condition number computation. Common matrix norms include the 1-norm (maximum column sum), infinity-norm (maximum row sum), and Frobenius norm (square root of sum of squared entries). The condition number is defined in terms of matrix norms.

113. Mean Value Theorem for Integration

The theorem stating that for a continuous function f on [a, b], there exists a point ξ in (a, b) such that ∫f(x)dx from a to b equals f(ξ)(b − a). It is used in deriving error bounds for numerical integration formulas by expressing the truncation error as f evaluated at some unknown interior point.

114. Midpoint Rule

A numerical integration formula that approximates the integral of f over [a, b] using the function value at the midpoint: ∫f(x)dx ≈ (b − a)·f((a + b)/2). The composite midpoint rule divides the interval into subintervals and applies this formula to each. It has second-order accuracy, the same as the trapezoidal rule.

115. Midpoint Runge-Kutta Method

A second-order Runge-Kutta method (also called RK2 or the midpoint method) that evaluates the slope at the midpoint of the time step: k₁ = hf(tₙ, yₙ), k₂ = hf(tₙ + h/2, yₙ + k₁/2), y_{n+1} = yₙ + k₂. It achieves second-order accuracy with two function evaluations per step.

116. Modified Euler Method

See Heun’s Method. A predictor-corrector second-order method that first uses Euler’s method to predict and then corrects using the average of slopes at the current and predicted points.

117. Monte Carlo Method

A numerical technique that uses random sampling to estimate mathematical quantities such as integrals or solutions to equations. For integration, random points are sampled within a domain and the fraction falling under the curve estimates the integral. Monte Carlo methods are especially useful in high-dimensional problems where deterministic methods are impractical.

118. Muller’s Method

A root-finding method that fits a quadratic polynomial through three successive approximations and uses the quadratic formula to find the next approximation. It converges with order approximately 1.84 (faster than secant but slower than Newton-Raphson) and can find complex roots without requiring complex starting points.

119. Multiple Roots

Roots of a function (or polynomial) at which the function and one or more of its derivatives are also zero. At a double root, f(x*) = 0 and f'(x*) = 0. Newton-Raphson converges only linearly at multiple roots rather than quadratically. Modified Newton-Raphson formulas can restore quadratic convergence near multiple roots.

120. Multi-Step Method

A numerical ODE method that uses solution values from two or more previous steps to compute the next value, as opposed to single-step methods which use only the current step. Multi-step methods are generally more efficient once started but require special starting procedures to generate the initial values needed.

121. Newton-Cotes Formulas

A family of numerical integration rules based on polynomial interpolation of the integrand at equally spaced nodes. The trapezoidal rule, Simpson’s 1/3 rule, and Simpson’s 3/8 rule are the most common Newton-Cotes formulas. They are derived by integrating the interpolating polynomial over the interval.

122. Newton-Raphson Method

An open root-finding method that uses the tangent line to the function at the current approximation to find the next approximation: x_{n+1} = xₙ − f(xₙ)/f'(xₙ). It converges quadratically near a simple root, making it the fastest standard root-finding method. It requires the derivative f'(x) and may diverge if the initial guess is poor or if f'(xₙ) is near zero.

123. Newton’s Backward Difference Interpolation

An interpolation formula using backward differences, preferred when interpolating near the end of a table of equally spaced data. It uses the backward difference operator ∇ and is particularly suited for extrapolation beyond the last tabulated value.

124. Newton’s Divided Difference Interpolation

An interpolation formula using divided differences, applicable to data points at unequal spacing. The polynomial is built recursively: P(x) = f[x₀] + fx₀,x₁ + fx₀,x₁,x₂(x−x₁) + … It is the most general form of Newton’s interpolating polynomial.

125. Newton’s Forward Difference Interpolation

An interpolation formula using forward differences, preferred when interpolating near the beginning of a table of equally spaced data. The formula is P(x) = f(x₀) + sΔf(x₀) + s(s−1)Δ²f(x₀)/2! + …, where s = (x − x₀)/h. It is one of the most frequently tested interpolation formulas on the board exam.

126. Newton’s Method for Systems

An extension of Newton-Raphson to systems of nonlinear equations, using the Jacobian matrix in place of the scalar derivative. Given a vector equation F(x) = 0, the iteration is x_{n+1} = xₙ − J(xₙ)⁻¹F(xₙ), where J is the Jacobian of F. In practice, the system J·Δx = −F is solved for the correction Δx at each step.

127. Nodal Analysis (Numerical)

The application of Kirchhoff’s current law at each node of an electrical network to produce a system of linear equations, solved numerically using Gaussian elimination or matrix inversion. It is the standard technique for analyzing electrical circuits numerically.

128. Node

In numerical interpolation, a node is one of the given data points xᵢ at which the function value is known. In finite difference and finite element methods, nodes are the discrete grid points where the solution is computed. The placement and spacing of nodes affects accuracy and stability.

129. Nonlinear Equation

An equation in which the unknown appears in a nonlinear form (squared, cubed, in a trigonometric function, etc.). Nonlinear equations generally cannot be solved analytically and require iterative numerical root-finding methods such as bisection, Newton-Raphson, or secant.

130. Nonlinear System

A system of equations in which at least one equation is nonlinear. Solving nonlinear systems requires iterative methods such as Newton’s method for systems, which uses the Jacobian matrix. The convergence analysis parallels that of the scalar Newton-Raphson method.

131. Norm

A function that assigns a non-negative magnitude to a vector or matrix. In numerical methods, norms are used to measure errors, residuals, and the size of iterates. The Euclidean (2-norm), maximum (infinity-norm), and 1-norm are commonly used for vectors. Convergence criteria are typically stated in terms of norms.

132. Numerical Differentiation

The approximation of derivatives using finite difference formulas. Forward, backward, and central difference formulas of various orders are used. Numerical differentiation amplifies round-off errors when the step size h is very small and introduces truncation errors when h is large, requiring a careful balance in selecting h.

133. Numerical Instability

The behavior of a numerical method in which errors grow unboundedly as computation proceeds, causing the computed solution to diverge from the true solution. Instability is usually triggered by a step size that is too large for the problem’s dynamics or by a method that is not suited to the equation’s stiffness.

134. Numerical Integration

The approximation of a definite integral using a weighted sum of function values at selected nodes. Also called numerical quadrature. Methods include Newton-Cotes formulas (trapezoidal rule, Simpson’s rule) and Gaussian quadrature. The accuracy depends on the smoothness of the integrand and the number of evaluation points.

135. Numerical Method

Any procedure that solves a mathematical problem using arithmetic operations on numbers rather than symbolic manipulation. Numerical methods produce approximate answers to problems that are difficult or impossible to solve analytically. The study of numerical methods includes algorithm design, error analysis, convergence, and computational efficiency.

136. Numerical Quadrature

See Numerical Integration. The evaluation of definite integrals using discrete summation formulas.

137. Numerical Stability

The property of a numerical method by which errors (from round-off, truncation, or initial data) do not grow disproportionately as computation proceeds. A stable method produces bounded errors throughout the computation. Stability is a necessary condition for the reliability of a numerical solution.

138. Open Integration Formula

A numerical integration rule that does not use function values at the endpoints of the integration interval. The midpoint rule is the simplest open Newton-Cotes formula. Open formulas are used when endpoint evaluation is impossible or undesirable.

139. Open Method

A root-finding method that does not require the root to be bracketed within an interval. Newton-Raphson, secant, and fixed-point iteration are open methods. They can converge much faster than bracketing methods but may diverge if the starting point is poorly chosen.

140. Order of Accuracy

A measure of how rapidly the error of a numerical method decreases as the step size h is reduced. A method of order p has error proportional to h^p. Doubling the number of steps (halving h) reduces the error by a factor of 2^p. Higher-order methods achieve greater accuracy with fewer steps.

141. Order of Convergence

A measure of how rapidly successive iterates approach the exact solution in an iterative root-finding method. For a method of order p, the error satisfies |e_{n+1}| ≈ C|eₙ|^p. Bisection is first order (p = 1), secant is approximately 1.618, and Newton-Raphson is second order (p = 2).

142. Overflow

A numerical error that occurs when a computation produces a value too large to be represented in the computer’s floating-point system. Overflow results in an infinite or undefined value and indicates that the computation has exceeded the representable range.

143. Partial Differential Equation (Numerical)

A differential equation involving partial derivatives, typically requiring finite difference, finite element, or finite volume methods for numerical solution over a spatial domain. PDEs arise in heat conduction, wave propagation, fluid flow, and electromagnetic field problems in engineering.

144. Partial Pivoting

A strategy in Gaussian elimination that selects the largest absolute value in the current column as the pivot, swapping rows as needed. Partial pivoting improves numerical stability by reducing the growth of round-off errors compared to natural (no pivoting) elimination.

145. Picard’s Iteration

An iterative method for solving initial value problems based on the integral form of the ODE. Starting from the initial function y₀(t) = y₀, successive approximations are computed as y_{n+1}(t) = y₀ + ∫f(s, yₙ(s)) ds. It is primarily a theoretical tool for proving existence and uniqueness of solutions and for deriving approximate analytical solutions.

146. Piecewise Linear Interpolation

Interpolation using straight-line segments connecting adjacent data points. It is equivalent to applying linear interpolation separately in each subinterval. While simple and robust, it only achieves first-order accuracy. Cubic spline interpolation is preferred when smoothness is required.

147. Pivot

The element used to eliminate other entries in Gaussian elimination. At each step, the pivot is the leading non-zero element in the current column (after possible row swapping for partial pivoting). A zero or near-zero pivot causes numerical difficulty and must be handled by pivoting.

148. Polynomial Interpolation

Interpolation using a polynomial that passes exactly through all given data points. The unique interpolating polynomial of degree at most n − 1 for n data points can be constructed using Lagrange’s formula, Newton’s divided difference formula, or other methods.

149. Power Method

An iterative method for finding the dominant eigenvalue and corresponding eigenvector of a matrix. Starting from an arbitrary non-zero vector, the method repeatedly multiplies by the matrix and normalizes, converging to the eigenvector of the largest eigenvalue. It is simple to implement but only finds one eigenvalue per application.

150. Predictor-Corrector Method

A two-phase ODE solving strategy that first uses an explicit formula (predictor) to estimate the next solution value and then applies an implicit formula (corrector) to improve the estimate using the predicted value. The Adams-Bashforth-Moulton method is the standard predictor-corrector pair. It combines the efficiency of explicit methods with improved accuracy.

151. Quadratic Convergence

A rate of convergence in which the number of correct decimal digits approximately doubles at each iteration. Formally, |e_{n+1}| ≈ C|eₙ|². Newton-Raphson exhibits quadratic convergence near simple roots. Quadratic convergence is much faster than linear convergence and is highly desirable in practice.

152. Quadratic Formula (Root Finding)

The exact formula for the roots of the quadratic equation ax² + bx + c = 0: x = (−b ± sqrt(b² − 4ac)) / (2a). While analytically exact, it can suffer from catastrophic cancellation in floating-point arithmetic when b² is much larger than 4ac, and reformulated versions are used for numerical stability.

153. Quadrature

See Numerical Integration. A term specifically referring to the numerical evaluation of a definite integral using a weighted sum of function values.

154. Quadrature Weights

The coefficients assigned to each function evaluation point in a numerical integration formula. In the trapezoidal rule, the weights are h/2 at the endpoints and h at interior points. In Gaussian quadrature, the weights are chosen to maximize the degree of precision. The sum of all weights equals the length of the integration interval.

155. Rectangular Rule

The simplest numerical integration formula, approximating the integral as the area of a rectangle whose height is the function value at one endpoint (left or right rectangle rule) or the midpoint. It has first-order accuracy for the endpoint versions and second-order for the midpoint version.

156. Reduced Row Echelon Form

The fully simplified form of an augmented matrix produced by Gauss-Jordan elimination, in which each pivot equals 1 and all other entries in pivot columns are zero. The reduced row echelon form directly reveals the solution without back substitution.

157. Regula Falsi Method

A bracketing root-finding method similar to bisection but using linear interpolation to select the next approximation instead of the midpoint. For a bracket [a, b], the new point is x = a − f(a)(b − a)/(f(b) − f(a)). It converges faster than bisection for well-behaved functions but can converge slowly when one endpoint moves very little. Also called the method of false position.

158. Relative Error

The ratio of the absolute error to the exact value (or the best available approximation): |approximate − exact| / |exact|. Relative error is dimensionless and meaningful when comparing errors across quantities of different magnitudes. It is more useful than absolute error when the scale of the answer is not known in advance.

159. Residual

The vector r = b − Ax, measuring how well a computed approximation x satisfies the linear system Ax = b. A small residual norm ‖r‖ indicates a good approximate solution if the system is well-conditioned. For ill-conditioned systems, a small residual can coexist with a large solution error.

160. Richardson Extrapolation

A technique for improving the accuracy of a numerical approximation by combining two approximations at step sizes h and h/2 to cancel the leading truncation error term. For a method of order p, the extrapolated result is (2^p · A(h/2) − A(h)) / (2^p − 1), which has accuracy of order p + 1 or higher.

161. Romberg Integration

A systematic application of Richardson extrapolation to the composite trapezoidal rule, building a triangular table of increasingly accurate estimates. Each column of the Romberg table corresponds to an order of accuracy one higher than the previous column. Romberg integration achieves high accuracy efficiently for smooth integrands.

162. Root

A value x* at which f(x*) = 0. Finding roots of nonlinear equations is one of the fundamental problems of numerical analysis. Roots may be real or complex, simple or multiple. Root-finding methods include bisection, Newton-Raphson, secant, regula falsi, and Muller’s method.

163. Root Finding

The numerical process of determining values of x that satisfy f(x) = 0. Root finding is one of the most common tasks in engineering computation and underlies the solution of nonlinear equations arising in circuit analysis, structural mechanics, thermodynamics, and control systems.

164. Round-Off Error

The error introduced by representing real numbers with a finite number of digits in floating-point arithmetic. Every arithmetic operation on a computer introduces a small round-off error. Accumulation of round-off errors over many operations can significantly degrade the accuracy of numerical results.

165. Runge-Kutta Methods

A family of explicit single-step methods for solving initial value problems that achieve high-order accuracy by evaluating the function f at multiple points within each step and computing a weighted average. The classical fourth-order Runge-Kutta method is the most widely used ODE solver in engineering practice and the most tested on the board exam.

166. Runge-Kutta-Fehlberg Method

An adaptive step size ODE solver that uses two Runge-Kutta formulas of different orders (typically 4th and 5th) to estimate the local truncation error and automatically adjust the step size to keep the error within a specified tolerance. It is the basis of the widely used RKF45 solver.

167. Runge’s Phenomenon

The tendency of high-degree polynomial interpolation at equally spaced nodes to exhibit large oscillations near the endpoints of the interpolation interval, even when the interpolated function is smooth. It demonstrates that increasing the degree of interpolation does not always improve accuracy and motivates the use of Chebyshev nodes or spline interpolation.

168. Secant Method

An open root-finding method that approximates the derivative in Newton’s formula using a finite difference between two previous iterates: x_{n+1} = xₙ − f(xₙ)(xₙ − x_{n−1})/(f(xₙ) − f(x_{n−1})). It avoids computing f'(x) directly and converges with order approximately 1.618 (the golden ratio). It requires two starting values.

169. Shooting Method

A technique for solving boundary value problems by converting them into initial value problems. An unknown initial condition is guessed, the IVP is solved, and the result at the final boundary is compared to the known boundary condition. The initial guess is adjusted iteratively (using bisection or Newton’s method) until the boundary condition is satisfied.

170. Significant Digits

The number of meaningful digits in a numerical value, indicating the precision of the measurement or computation. Numerical methods aim to produce results with as many significant digits as the underlying data and arithmetic precision allow. Loss of significant digits occurs through cancellation, round-off, and ill-conditioning.

171. Simpson’s 1/3 Rule

A Newton-Cotes integration formula using three equally spaced points (two subintervals): ∫f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + f(x₂)], where h = (x₂ − x₀)/2. It fits a quadratic polynomial through the three points and integrates it exactly. It has fourth-order accuracy (degree of precision 3) and is one of the most tested integration formulas on the board exam.

172. Simpson’s 3/8 Rule

A Newton-Cotes integration formula using four equally spaced points (three subintervals): ∫f(x)dx ≈ (3h/8)[f(x₀) + 3f(x₁) + 3f(x₂) + f(x₃)], where h = (x₃ − x₀)/3. It fits a cubic polynomial through the four points. It has the same fourth-order accuracy as Simpson’s 1/3 rule but is used when the number of subintervals is a multiple of 3.

173. Single-Step Method

A numerical ODE method that computes the solution at step n + 1 using only the solution at step n, without storing or using information from earlier steps. Euler’s method, Heun’s method, and all Runge-Kutta methods are single-step methods. They are self-starting but may be less efficient than multi-step methods for the same accuracy.

174. Sparse Matrix

A matrix in which most entries are zero. Sparse matrices arise naturally in finite difference and finite element methods, where each grid point is connected only to its immediate neighbors. Specialized sparse storage formats and solution algorithms exploit sparsity to reduce memory and computation requirements dramatically.

175. Spline

A piecewise polynomial function constructed by connecting polynomial segments at specified breakpoints (knots), with continuity conditions on the function and its derivatives at the joins. Cubic splines are the most common, providing a smooth interpolant with continuous first and second derivatives at all interior knots.

176. Stability

See Numerical Stability. The property of a numerical method that prevents the unbounded growth of errors during computation.

177. Starting Procedure

The initialization required for multi-step ODE methods, which cannot start from a single initial condition because they need values at multiple previous steps. Single-step methods (typically Runge-Kutta) are used to generate the first few solution values needed to start the multi-step method.

178. Stiff Equation

A differential equation for which certain explicit numerical methods require a very small step size to maintain stability, even though the solution itself may be smooth and slowly varying. Stiffness typically arises from widely separated time scales in the equation. Implicit methods are preferred for stiff problems.

179. Stopping Criterion

See Convergence Criterion. The condition used to terminate an iterative computation when the solution is deemed sufficiently accurate.

180. Subinterval

One of the smaller intervals into which the integration interval [a, b] is divided in a composite numerical integration rule. Each subinterval has width h = (b − a)/n for n equal subintervals. Increasing the number of subintervals improves accuracy by reducing the width h.

181. Successive Approximation

The iterative process of generating a sequence of approximations x₀, x₁, x₂, … that converge to the exact solution. Fixed-point iteration is a specific form of successive approximation. The method is used for root finding, solving integral equations, and solving differential equations.

182. Superlinear Convergence

A rate of convergence faster than linear but not as fast as quadratic. The secant method converges superlinearly with order approximately 1.618. Superlinear convergence is characterized by |e_{n+1}| / |eₙ|^p → C for some 1 < p < 2.

183. Synthetic Division

An efficient algorithm for dividing a polynomial by a linear factor (x − r), producing the quotient polynomial and remainder. It is used in Horner’s method for polynomial evaluation, in polynomial deflation after finding a root, and in Bairstow’s method. The remainder equals the polynomial value at x = r.

184. Taylor Series Method

An ODE solver that uses higher-order terms of the Taylor series expansion of the solution to achieve high-order accuracy: y_{n+1} = yₙ + hy’ₙ + (h²/2)y”ₙ + …. It requires computation of higher derivatives of f, which limits its practical use. Runge-Kutta methods were developed to achieve high-order accuracy without computing higher derivatives.

185. Tolerance

A user-specified bound on the acceptable error in a numerical computation. Iterative methods terminate when successive iterates differ by less than the tolerance or when the function value falls below the tolerance. The tolerance must balance the competing demands of accuracy and computational cost.

186. Trapezoidal Rule

A Newton-Cotes integration formula approximating the integral by the area of a trapezoid: ∫f(x)dx ≈ (h/2)[f(a) + f(b)], where h = b − a. It is derived by fitting a linear polynomial to the two endpoints. The trapezoidal rule has second-order accuracy and is exact for linear functions. The composite trapezoidal rule applies it over multiple subintervals.

187. Triangular System

A system of linear equations represented by an upper or lower triangular matrix. Triangular systems are solved directly by back substitution (upper triangular) or forward substitution (lower triangular). Gaussian elimination transforms a general linear system into upper triangular form.

188. Truncation Error

The error introduced by replacing an exact mathematical operation (such as a derivative or integral) with a finite approximation (such as a finite difference or quadrature rule). Truncation error arises from truncating an infinite series and decreases as step size h decreases. It is the primary source of error in well-implemented numerical methods.

189. Two-Point Boundary Value Problem

A boundary value problem in which conditions are specified at exactly two points (typically the endpoints of an interval). The shooting method and finite difference method are the standard numerical approaches for two-point BVPs.

190. Underflow

A numerical error that occurs when a computation produces a value too small to be represented as a non-zero floating-point number. Underflow typically results in the value being set to zero, which may cause subsequent errors if the value is used in division or as a denominator.

191. Unimodal Function

A function with exactly one local maximum or minimum in a given interval. Unimodal functions are important in numerical optimization because methods like golden section search are guaranteed to find the optimal value within the interval. Optimization algorithms often require the objective function to be unimodal within the search interval.

192. Uniqueness of Interpolating Polynomial

The theorem stating that through any n + 1 distinct data points, there is exactly one polynomial of degree at most n that passes through all of them. While the interpolating polynomial is unique, it can be expressed in many different forms (Lagrange, Newton, etc.) that are all equivalent.

193. Upper Triangular Matrix

A square matrix in which all entries below the main diagonal are zero. The result of Gaussian elimination without pivoting is an upper triangular matrix. Systems with upper triangular coefficient matrices are solved by back substitution.

194. Vector Norm

A scalar measure of the size or length of a vector, used in convergence criteria and error analysis. The Euclidean norm ‖x‖₂ = sqrt(Σxᵢ²), the infinity norm ‖x‖∞ = max|xᵢ|, and the 1-norm ‖x‖₁ = Σ|xᵢ| are the most common. Convergence of iterative methods is typically measured by the norm of the difference between successive iterates.

195. Weddle’s Rule

A Newton-Cotes numerical integration formula using seven equally spaced points (six subintervals). It has sixth-order accuracy and is occasionally used when Simpson’s rule is insufficient. The formula gives weights in the pattern 3/10 times h times [1, 5, 1, 6, 1, 5, 1] across the seven points.

196. Weighted Residual Method

A broad class of methods for solving differential equations by requiring the residual (error in satisfying the equation) to be zero in a weighted average sense. The finite element method and Galerkin method are weighted residual methods. The choice of weight function determines the specific method.

197. Well-Conditioned Problem

A problem in which small changes in the input data produce small, proportional changes in the solution. Well-conditioned problems have small condition numbers and are reliably solved by standard numerical methods. Most engineering problems are well-conditioned near their expected operating range.

198. Well-Posed Problem

A mathematical problem that has a solution, the solution is unique, and the solution depends continuously on the initial data. Well-posedness is a prerequisite for reliable numerical solution. Ill-posed problems require specialized regularization techniques.

199. Zero of a Polynomial

A root of a polynomial equation p(x) = 0. By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots in the complex number system, counted with multiplicity. Numerical methods for polynomial roots include Newton-Raphson, Bairstow’s method, and Muller’s method. Deflation is used to reduce the polynomial after each root is found.

200. Zero-Stability

A property of multi-step numerical ODE methods ensuring that small perturbations in initial data do not cause unbounded growth in the computed solution as the step size approaches zero. Zero-stability is a necessary condition for convergence and is satisfied by consistent multi-step methods with roots of the characteristic polynomial satisfying the root condition.

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201. Zeros of a Function

The values of x for which f(x) = 0. Also called roots. Finding zeros is the fundamental objective of root-finding numerical methods. Engineering problems involving equilibrium conditions, resonance frequencies, and critical points all reduce to finding zeros of appropriate functions.

CONCLUSION

For the engineering Mathematics board exam, the highest-priority items in Numerical Methods are the root-finding methods particularly bisection and Newton-Raphson and the numerical integration formulas, especially the composite trapezoidal rule and Simpson’s 1/3 rule. These are the topics that appear most consistently across exam years and across engineering disciplines. You should be able to execute several iterations of bisection, apply one or two Newton-Raphson steps from a given starting point, and evaluate composite Simpson’s and trapezoidal rules from a data table without hesitation. The formulas must be memorized and practiced until they are automatic.

After the root-finding and integration formulas, focus on Euler’s method and the fourth-order Runge-Kutta method for initial value problems. The board exam tests your ability to carry out one or two steps of these algorithms by hand given f(t, y), an initial condition, and a step size. For Euler’s method, the formula is simple but the concept of accumulated error and stability is important. For the classical Runge-Kutta method, memorize the four slope evaluations k₁ through k₄ and the weighted average formula. Interpolation is also testable; Newton’s forward difference formula and Lagrange interpolation are the two forms you should know how to apply.

Finally, connect the vocabulary of error analysis to the methods you know. Every board exam question on numerical methods either asks you to apply a method or asks you to identify a property such as the order of accuracy, the convergence rate, the type of error, or the condition under which a method is valid. The definitions in this glossary are the direct preparation for those identification questions. Review this glossary alongside working practice problems, and pay particular attention to the conditions and limitations of each method on when it converges, how fast, and what can cause it to fail. That combination of computational skill and conceptual clarity is exactly what the licensure examination rewards.

For practice problems on all these topics, head over to our Advanced Mathematics Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.