Simpson's rule approximates the integral by fitting three points of the function to a parabola and then integrating that parabola. It is exact for constant, linear, and quadratic functions.
Conditions: Uses three points of the function on the integration interval.; Exact for constant, linear, and quadratic functions according to the lecture statement.
Simpson's rule approximates the integral by fitting three points of the function to a parabola and then integrating that parabola. It is exact for constant, linear, and quadratic functions.
Conditions: Uses three points of the function on the integration interval.; Exact for constant, linear, and quadratic functions according to the lecture statement.
The trapezoidal rule approximates an integral by replacing the curve over an interval with a straight line between two sampled points and computing the area of the resulting trapezoid. In composite form, the endpoint values are each weighted by one half because interior points belong to two neighboring trapezoids, while endpoints belong to only one.
Conditions: Applied to numerical integration over an interval [a,b].; Uses function values at the interval endpoints.; For composite use, adjacent subinterval endpoints are shared, producing half weights at the outer endpoints.
The trapezoidal rule approximates an integral by replacing the curve over an interval with a straight line between two sampled points and computing the area of the resulting trapezoid. In composite form, the endpoint values are each weighted by one half because interior points belong to two neighboring trapezoids, while endpoints belong to only one.
Conditions: Applied to numerical integration over an interval [a,b].; Uses function values at the interval endpoints.; For composite use, adjacent subinterval endpoints are shared, producing half weights at the outer endpoints.
Halving each interval is recommended because it allows previous function values to be reused in the next computation, reducing redundant evaluations of the function, which often dominate the runtime.
Conditions: Refinement is by successive bisection of existing subintervals.; Benefit comes from reusing previously computed function values.
Halving each interval is recommended because it allows previous function values to be reused in the next computation, reducing redundant evaluations of the function, which often dominate the runtime.
Conditions: Refinement is by successive bisection of existing subintervals.; Benefit comes from reusing previously computed function values.