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The midpoint method computes + so that the red chord is approximately parallel to the tangent line at the midpoint (the green line). In numerical analysis , a branch of applied mathematics , the midpoint method is a one-step method for numerically solving the differential equation ,
If = (+) / for all i, the method is the midpoint rule [2] [3] and gives a middle Riemann sum. If f ( x i ∗ ) = sup f ( [ x i − 1 , x i ] ) {\displaystyle f(x_{i}^{*})=\sup f([x_{i-1},x_{i}])} (that is, the supremum of f {\textstyle f} over [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]} ), the method is the upper rule and gives an upper ...
Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction.The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the ...
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
The y arc elasticity of x is defined as: , = % % where the percentage change in going from point 1 to point 2 is usually calculated relative to the midpoint: % = (+) /; % = (+) /. The use of the midpoint arc elasticity formula (with the midpoint used for the base of the change, rather than the initial point (x 1, y 1) which is used in almost all other contexts for calculating percentages) was ...
In computer graphics, the midpoint circle algorithm is an algorithm used to determine the points needed for rasterizing a circle. It is a generalization of Bresenham's line algorithm . The algorithm can be further generalized to conic sections .
Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist.
In geometry, the sagitta (sometimes abbreviated as sag [1]) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. [2] It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror ...