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Mathematics 30 flashcards ~15 min

Numerical Methods and Approximation

Numerical methods provide practical tools for approximating solutions when exact analytical answers aren't feasible. This deck covers root-finding techniques like the bisection, Newton-Raphson, and secant methods, along with numerical integration approaches su...

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Numerical methods provide practical tools for approximating solutions when exact analytical answers aren't feasible. This deck covers root-finding techniques like the bisection, Newton-Raphson, and secant methods, along with numerical integration approaches such as the trapezoidal rule and Simpson's rule. You'll also review error analysis concepts including round-off and truncation error, convergence and stability, and foundational tools like Euler's method, Gaussian elimination, and the Monte Carlo method. Ideal for engineering, computer science, or applied math students studying computational approximation techniques.

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A technique for approximating solutions to mathematical problems using numbers.
To solve problems that lack exact analytical solutions.
Finding roots of a continuous function by repeatedly halving an interval.
The function must change sign over the interval.
Finding successively better approximations of a function's roots.
The function's derivative.
Estimating a value between two known values using a straight line.
Estimating values outside the range of known data.
Approximating the area under a curve (numerical integration).
A more accurate method of numerical integration using parabolic segments.
The difference caused by rounding numbers during computation.
The error from approximating a mathematical process by a finite number of steps.
A method that repeats a process to get closer to a solution each time.
When successive approximations get closer to the true solution.
When successive approximations move away from the true solution.
Approximating solutions to ordinary differential equations.
A root-finding method using a sequence of secant lines.
A method that finds solutions by repeatedly applying a function.
Estimating a derivative using values of a function at nearby points.
A technique approximating derivatives using differences between function values.
A starting estimate used to begin the approximation process.
The magnitude of the difference between an approximate and exact value.
The absolute error divided by the exact value.
A measure of how sensitive a function's output is to small input changes.
Solving systems of linear equations.
Factoring a matrix into lower and upper triangular matrices.
Estimating unknown values that fall within known data points.
Approximating functions as an infinite sum of polynomial terms.
The property that small errors do not grow uncontrollably during computation.
A technique using random sampling to approximate numerical results.