Quotient difference

Mathematical concept

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The difference quotient is the mean difference, and the first order difference quotient is the approximate value of the first derivative. reciprocity step (h) Of Discrete function F (x), its n-order difference quotient is the ratio of its n-order difference to the n-power of its step size. For example, when n=1, if the difference is forward or backward, the first order difference quotient obtained is the first order approximation of the derivative of the function; If the difference is centered, the first order difference quotient obtained is the second order approximation of the derivative.^[1]

Chinese name: Quotient difference
Foreign name: difference quotient

Discipline: mathematics
Related concepts: Interpolation, interpolation node, difference quotient table, etc
Type: Mathematical concept

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definition

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Given function

and interpolation node

use

express

About nodes

K-th Difference Quotient (k=1, 2,..., n), which can be defined recursively as

among

About nodes

0 order difference quotient of is defined as its function value, that is

。^[2]

Difference quotient table

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The difference quotient table (Table 1) can be constructed and calculated recursively by column

Difference quotient of each order of:^[2]

Table 1 Calculation of Difference Quotient
node	Difference quotient of order 0	First order difference quotient	Second order difference quotient	3rd order difference quotient	Fourth order difference quotient