# Numerical Analysis problem?

Suppose that S(f,h) is a quadrature rule for the integral I=integral_a^bf(x)dx and that the error series is (c4)*h^4+(c6)*h^6+.... Combine S(f,h) with S(f,h/3) to find a more accurate approximaton to I

Relevance

I = S(f,h) + (c4)*h^4 + (c6)*h^6 + ...

I = S(f,h/3) + (c4)*h^4/81 + (c6)*h^6 /3^6 + ...

81*I = 81 S(f,h/3) + (c4)*h^4 + (c6)*h^6/9 + ...

80 I = 81 S(f,h/3) -- S(f,h) -- (8/9) (c6)*h^6 +...

I = [(81/80) S(f,h/3) -- S(f,h) /80] -- (c6)*h^6 /90 +...

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• The Rhind Mathematical Papyrus is a replica from 1650 BCE of an impressive until eventually now artwork and shows us how the Egyptians extracted sq. roots. In historic India, the actuality of theoretical and utilized aspects of sq. and sq. root replaced right into a minimum of as late to very actuality the Sulba Sutras, dated round 800-500 B.C. (probably a lot until eventually now). a way for learning astonishing approximations to the sq. roots of two and three are given contained in the Baudhayana Sulba Sutra. Aryabhata contained in the Aryabhatiya (area 2.4), has given a way for learning the sq. root of numbers having many digits. D.E. Smith in historic past of mathematics, says, on the topic of the contemporary difficulty in Europe: "In Europe those procedures (for learning out the sq. and sq. root) did now no longer seem until eventually now Cataneo (1546). He gave the approach of Aryabhata for determining the sq. root".