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The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated. Interpolating two values yields a line: a polynomial of degree one. This is the basis of the secant method. Regula falsi is also an interpolation method that interpolates two points at a time but it differs ...
CORDIC (coordinate rotation digital computer), Volder's algorithm, Digit-by-digit method, Circular CORDIC (Jack E. Volder), [1] [2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), [3] [4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), [5] [6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots ...
The locus of points that give the same value in the algorithm, for different values of alpha and beta. The alpha max plus beta min algorithm [1] is a high-speed approximation of the square root of the sum of two squares.
To reduce this error, one may, for each root that is found, restart Newton's method with the original polynomial, and this approximate root as starting value. However, there is no warranty that this will allow finding all roots. In fact, the problem of finding the roots of a polynomial from its coefficients can be highly ill-conditioned.
Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)
The simplest form of the formula for Steffensen's method occurs when it is used to find a zero of a real function ; that is, to find the real value that satisfies = .