Numerical Modeling Lecture 1-12

Coefficient matrix

Square array containing the coefficients of the unknowns in a system of linear equations

Runge phenomenon

Oscillatory behavior that occurs when high-degree polynomial interpolation is applied over equally spaced points

NaN (Not a Number)

Special floating-point value indicating an undefined or unrepresentable result, such as 0/0

Simpson’s Rule

An integration technique that fits quadratic polynomials to subintervals to improve accuracy over the trapezoidal rule

Iterative refinement

A technique that repeatedly improves an approximate solution by correcting residual errors

Polynomial Interpolation

Construction of a polynomial that passes exactly through a given set of data points

Error bound for Taylor polynomial

Expression that limits the remainder term using the next derivative evaluated at a intermediate point

Stability of an algorithm

Property describing how errors are amplified or damped as the algorithm progresses

Precision loss due to scaling

Reduction in significant digits when a value is multiplied or divided by a large factor before computation

Mantissa (significand)

23-bit fraction that, together with the exponent, determines the magnitude of the value

Determinant

Scalar value computed from a square matrix that indicates whether the matrix is invertible

Scaling for numerical stability

Adjusting the magnitude of inputs or intermediate results to avoid loss of significance during subtraction of similar numbers

Error propagation analysis

Method for estimating how uncertainties in inputs affect the uncertainty of the final output

Floating-point representation of zero

All bits zero for +0 and sign bit set with all other bits zero for -0, both considered equal in magnitude

3×3 matrix

Square array with three rows and three columns, often used to represent three simultaneous linear equations

Numerical solution

Approximate result obtained by applying computational algorithms to discretized equations

Natural logarithm Taylor series

Series: ln(1+x) = x − x²/2 + x³/3 −…
Valid for |x| < 1

Bit precision

Number of binary digits used to store a value, directly affecting the smallest distinguishable change

Floating-point rounding modes

Options such as round-to-nearest, round-toward-zero, round-up, and round-down that dictate how results are truncated

IEEE 754

A standard that defines binary floating-point formats, rounding, rules, and special values for computers

Argument reduction

A technique that maps a large input to an equivalent value within a primary interval to improve computational accuracy

Linear system

Set of equations in which each terms is either a constant or a product of a constant and a single variable

Normalization of scientific notation

Adjusting a number so that its leading digit is non-zero, analogous to the implicit leading 1 in binary floating-point

Pivoting in Gaussian elimination

Reordering rows or columns to place larger elements on the diagonal, reducing round-off error

Floating-point overflow handling

A procedure that substitutes infinity or the largest representable value when a computation exceeds the upper limits

LU (lower-upper) decomposition

Factorization of a matrix into a lower-triangular and an upper-triangular matrix to simplify solving linear systems

Cosine Taylor series

Series cos x = 1 − x²/2! + x^4/4! − …
that converges for all x

Subtracting nearly equal numbers

Operation that can cause catastrophic cancellation, dramatically reducing the number of accurate digits

Exponent field

8-bit section that stores a biased exponent to scale the mantissa

Relative error

Ratio of the absolute error to the true value, often expressed as a percentage

Known vector

Column vector representing the constant terms on the right-hand side of a linear system

Rounding error

Difference between the exact mathematical result and its rounded floating-point representation

Newton-Raphson method

Iterative root-finding technique that use4s the function and its derivative to converge quadratically to a zero

Adaptive step size

Strategy that varies the interval size during integration to maintain a desired error tolerance

Binary exponent bias for 32-bit format

Constant value 127 added to the actual exponent to encode it as an unsigned 8-bit field

Machine epsilon

The smallest number that, when added to 1.0, yields a distinguishable result in the given floating-point format

Underflow

Condition where a number is too small to be represented in normalized form, causing it to become zero

Floating-point exponent bias

Constant added to the actual exponent to allow representation of both positive and negative exponents

Denormalized number

Floating-point value with a leading mantissa bit of 0, used to represent numbers closer to zero than the smallest normalized value

Cramer’s Rule

Method for solving a linear system by expressing each variable as a ratio of two determinants

Euler (historical contribution)

Pioneered numerical integration formulas and series expansions for differential equations

Space matrix

Matrix in which most elements are zero, allowing specialized storage and faster operations

Convergence testing

Procedure that checks whether successive refinements of a computation approach a stable result

Analytical solution

Exact expression derived from mathematical theory that gives a closed-form result

Floating-point underflow handling

Procedure that substitutes zero or a denormalized value when a computation falls below the smallest normalized magnitude

Trapezoidal rule

Numerical integration method that approximates the area under a curve by summing areas of trapezoids

Unknown vector

Column vector that holds the variables to be solved in a linear system

Condition number

A metric that quantifies the sensitivity of a system’s solution to small changes in the input data

Small-angle approximation

Simplification where sinθ ≈ θ and tanθ ≈ θ when θ is measured in radians and is near zero

Convergence rate

Speed at which an iterative method approaches the exact solution as the step size or iteration count increases

Jacobian matrix

Matrix of first-order partial derivatives that describes how a vector-valued function changes near a point

Laplace expansion

Recursive formula for determinant calculation that expands along a chosen row or column using minors and cofactors

Discrete Fourier Transform (DFT)

Algorithm that converts a finite sequence of equally spaced samples into frequency components

Lagrange form of interpolation polynomial

Explicit formula that combines basis polynomials weighted by the funciton values at the nodes

Normalized mantissa implicit bit

Assumed leading 1 in the mantissa of a normalized floating-point number, which is not stored explicitly

Arbitrary precision arithmetic

Software technique that allows calculations with a user-defined number of digits beyond hardware limits

Sign bit

Single binary digit that indicates whether the store number is positive (0) or negative (1)

Gauss (historical contribution)

Introduced systematic elimination methods and contributed to the theory of least squares

Absolute error

Magnitude of the difference between an approximate value and the exact mathematical value

32-bit floating-point format

Binary representation using 1 sign bit, 8 exponent bits, and 23 mantissa bits

Unit scaling

Choosing measurement units that keep numerical values within the optimal range of the floating-point format

Normalized number

Floating-point value whose leading mantissa bit is implicitly 1, providing maximum precision

Round-to-nearest even

Tie-breaking rule that rounds a value to the nearest representable number; if exactly halfway, selects the one with an even least-significant bit

Finite difference approximation

Method that estimates derivatives by using function values at discrete points

Taylor series

Infinite sum of derivatives evaluated at a point, used to approximate smooth function locally

Overflow

Condition where a number exceeds the largest representable magnitude, resulting in infinity

Newton (historical contribution)

Developed early numerical techniques such as the Newton-Raphson method for root finding