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enter image description here The picture is a extract of Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun.

What is the meaning of $m$ in this table and how $DA_k^n(m)$ is related to $A_k(m)$ ?

Take for example $n=5$ . I know that $\int_{x}^{x+1}f(x)dx= h(\frac{251}{720}f_{m+1}+\frac{646}{720}f_{m}-\frac{264}{720}f_{m-1}+\frac{106 }{720}f_{m-2}-\frac{19}{720}f_{m-3})$

But for $n=5$ there are two row ,-2 and -1, which one should we take?

amilton moreira
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1 Answers1

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Actually, what you get is $$\int_{x_{-2}}^{x_{-1}}f(x)dx=h\left\{\frac{251}{720}f(x_{-2})+\frac{646}{720}f(x_{-1})-\frac{264}{720}f(x_0)+\frac{106}{720}f(x_1)-\frac{19}{720}f(x_2)\right\}$$ This is for integrating between the first (or last) two entries of a table of equally-spaced data. The next row says $$\int_{x_{-1}}^{x_0}f(x)dx=h\left\{-\frac{19}{720}f(x_{-2})+\frac{346}{720}f(x_{-1})+\frac{456}{720}f(x_0)-\frac{74}{720}f(x_1)+\frac{11}{720}f(x_2)\right\}$$ And is to be preferred for table entries in the middle of the table because at least the roundoff error and maybe also the truncation error are smaller when this formula is applicable. Here's a Matlab program that can spit out this table using the method of undetermined coefficients:

% M_I.m

fudge = [1/2 1 3/2 1];
for k = 3:2:9,
    D = factorial(k+1)*fudge((k-1)/2)
    for m = 0:-1:(3-k)/2,
        A = (ones(k,1)*[m:k-1+m]).^([0:k-1]'*ones(1,k));
        b = 1./[1:k]';
        x = round(D*(A\b)')
    end
end
user5713492
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  • Could you give a reference that explain this well? – amilton moreira Feb 28 '20 at 20:53
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    My reference would be Abramowitz & Stegun, but I guess that would be cyclic in this context. Assuming you are taking an undergraduate numerical methods class and have the student edition of Matlab (should be required for the class) you can pick apart my program and figure out how it works, thus acquiring more insight into the problem. – user5713492 Feb 28 '20 at 21:24