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Translation:Methodus nova integralium valores per approximationem inveniendi

From Wikisource
New Method for Finding the Values of Integrals by Approximation (1814)
by Carl Friedrich Gauss, translated from Latin by Wikisource
Carl Friedrich Gauss4442186New Method for Finding the Values of Integrals by Approximation1814Wikisource


1.

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Among the methods proposed for the numerical approximation of integrals, a prominent place is held by the rules which were developed by Newton and refined by Cotes. Specifically, if the value of the integral taken from to is required, then the values of for these limiting values of and for several other intermediate values progressing by equal increments from first to last, are to be multiplied by certain numerical coefficients. This being done, the sum of the products, multiplied by will supply the desired integral, with greater precision as more terms are used in this operation. Since the principles of this method, which seems to be called into use less frequently by geometers than it should, have nowhere, as far as I know, been fully explained, it will not be out of place to say a few things about them.

2.

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Let us agree to use a multitude of terms, and let so that the values of are etc. up to and correspondingly the values of are etc. up to finally, let so that can also be regarded as a function of Let represent the function

or where represents each of the integers , , ,


and is the value of for

It is clear that represents an integral algebraic function of of order and its values for each of the values of namely are equal to the values of It is also clear that if is another integral function producing the same values of for the same values of then will vanish for the same values, and therefore it must be divisible by the factors and therefore also by their product (which is of order ), from which it is clear that must, unless it is identical to be of a higher order, meaning that is the only integral function among those not exceeding order which coincides with for those values. Therefore, if , when expanded into a series of powers of , breaks off before the term involving , it will be identical to and if the series converges so quickly as to allow the subsequent terms to be neglected, then the function can replace within the limits or .

3.

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Now our integral is transformed into taken from to and as we have just indicated, we will replace this with Thus by expanding into

the integral from to will be

and setting this quantity the desired integral will be

For example, let us compute the coefficient for Here we have

Hence and therefore

The computation can be shortened a bit by setting Then we have

Let us set

where the numerator should end in if is odd, or in if is even. Then

Now the integral taken from to is equal to the integral

from to

Therefore, by setting

(it being evident that the powers etc. are absent), the first part of the integral will vanish for odd values of while the other part will vanish for even values, so that the integral becomes

for even values of and

for odd values of In our example we have

hence
as above.

It is worth noting that and therefore with the upper sign holding for even and the lower sign for odd Hence, since it is easy to see that we will always have meaning that the last coefficient is equal to the first, the penultimate to the second, and so on.

4.

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We hereby append the numerical values of these coefficients, up to which were computed by Cotes in Harmonia Mensurarum.

For or two terms.

For or three terms.

For or four terms.

For or five terms.

For or six terms.

For or seven terms.

For or eight terms.

For or nine terms.

For or ten terms.

For or eleven terms.

5.

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Since the formula exactly represents the integral from to or the integral from to whenever the expansion of into a series does not go beyond the power , but otherwise only approximates it, it remains to show how to account for the error induced by the immediately following terms. Let us denote generally by the difference between the true value of the integral from to and the value derived from the formula. Then

etc. It is evident, therefore, that if is expanded into a series

then the difference between the true value of the integral and the approximated value derived from the formula can be expressed as

But evidently, , etc. up to are all automatically thus, the correction of the approximated formula will be

The nature of quantities , etc. will be examined more accurately later; here, it suffices to provide the numerical values of the first or second, for each value of so that the degree of precision afforded by the approximate formula can be estimated.

For , we have
For , we find
For , it is
For
For
For
For
For
For
For


For all even values of here, we observe and furthermore for odd values of , however, we always have The reason why this occurs can be deduced easily from the following considerations.

In general, let denote the difference between the true value of the integral from to and the value derived from the approximate formula, so that we have

with the integral being taken from to . Clearly, for odd values of both the true and approximate integral values vanish: hence , etc., and generally for all odd values of For even values of on the other hand, the formula can be written as

if is even; or

if is odd.

Therefore, if expanding in a series according to powers of yields

then the correction to be applied to the approximate value of the integral from to will be

or rather, since necessarily vanishes for any integer value of no greater than the correction will be

for even or

for odd

The corrections can be easily converted to and vice versa. For if we have

then

And similarly,

The terms where is affected by an odd index will be eliminated from the latter formula, and each should only be continued up to the index (inclusive). Therefore, it is clear that we will have

for even

for odd

from which the above observations can be deduced.

6.

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Generally speaking, it will therefore be preferable to assign an even value to or to employ an odd number of terms, when applying the method of Cotes. Indeed, very little precision will be gained by ascending from an even value of to the next highest odd one, as the error remains of the same order, although affected by a slightly smaller coefficient. Conversely, ascending from an odd value of to the next highest even one will increase the order of the error by two, and the coefficient being significantly reduced, so the precision will increase. So if five terms are used, that is, for the error is approximately expressed by or if we set the error will be approximately or thus it will not even be half of the former: on the other hand, for the error becomes approximately or and the precision is increased all the more, as the series into which the function has been expanded converges more quickly.

7.

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Following these preliminaries regarding the method of Cotes, we proceed to a general inquiry, discarding the condition that the values of progress in an arithmetic progression. We thus address the problem of determining the value of the integral between given limits from some given values of either exactly or as closely as possible. Let us assume that the integral is to be taken from to and let us introduce another variable so that the integral from to needs to be investigated.

Let be distinct values of let the corresponding values of be and let denote the following integral algebraic function of order

If is set equal to any of the quantities , it is clear that the values of this function, coincide with the corresponding values of the function from which, as concluded in art. 2, we deduce that is identical to , provided that is also an integral algebraic function of order no greater than or at least it can take the place of if can be converted into a series of powers of which exhibits such convergence that it is permissible to neglect the higher order terms.

8.

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To evaluate the integral let us consider each part of separately. Let denote the product

and through the expansion of this product, let

The numerator of the fraction by which is multiplied in its respective part of becomes the numerators in the subsequent parts are likewise , etc. The denominators are nothing but the values determined by these numerators if is set respectively to , etc. Let us denote these denominators respectively by , etc., so that we have

When for we have the identical equation

and therefore

Thus, dividing by , we get

The value of this function for is obtained as

Hence is equal to the value of for as is evident for other reasons. Similarly, etc. will be the values of for , etc.

Furthermore, we find the value of the integral from to to be:

Let us arrange these terms in the following order:

It is clear that the same quantity arises if, in the product obtained by multiplying the function by the infinite series

all terms involving negative powers of are rejected (or in short, in the integral part of the product, which is an integral function of ), is replaced by Therefore let us set [1]

so that is the integral function of contained in this product, and is the other part, namely the series descending to negative infinity. Then the value of the integral from to will be equal to the value of the function at So, if we denote the values of the function

determined by , etc., up to resp. by then the integral from to will be

which, when multiplied by , will give the value, either exact or approximate, of the integral from to .

9.

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These operations are somewhat easier to perform if we introduce another variable For the sake of brevity, we also write etc. By substituting the value for let be transformed into or equivalently let

Then and hence etc. are the values of determined by etc.

Since the series etc. is nothing but substituting will transform it into etc. Therefore, if we set

so that is the integral function of contained in this product, and is the other part, which is an infinite descending series, it is clear that

However, it is clear that , being an integral function of will necessarily become an integral function of as a result of the substitution on the other hand, , which contains only negative powers of will only generate negative powers of as a result of the same substitution. Therefore, will be nothing but transformed by this substitution, and likewise will be produced from Consequently, it makes no difference whether we substitute into or into From this we conclude that etc. are also the values of the function determined by etc.

10.

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Before we proceed further, we will illustrate these precepts with an example. Let and suppose that Then we have

Multiplying by etc., we obtain

Hence the values of the coefficients are expressed by the fractional function

wherein the values are subsequently substituted for The other method, which is a bit faster, yields

from which , etc. will be values of the fractional function

for , etc. Both methods yield the same numbers given in Art. 4 of Harmonia Mensurarum. However, in such an example like this, where , etc. are all rational quantities, the values of the denominator are more conveniently computed in the original form, namely for and likewise for the others. The same holds for the denominator , which for becomes

11.

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When , etc., are either partially or altogether irrational, it will be useful to transform the fractional function, from which we derive the numbers , etc., into an integral function. Since an elementary explanation of this transformation cannot be found in algebraic books, we will provide one here. Specifically, let be three indeterminate integral functions of and let us seek an integral function which can be substituted for the fraction , as far as is taken to be any root of the equation Let us assume that does not vanish for any of these values of , or equivalently, that and imply no common indeterminate divisor. We will denote the exponents of the highest powers of in and by respectively.

Divide by as is usual, until the order of the remainder is less than let the remainder be and let its order be so that is the highest order term of the remainder; we will denote the quotient of this division by Similarly, divide the function by let the residue of order be obtained as then again from the division of the function by let the residue of order be obtained as , and so on, until in the series of functions etc., each having its highest term affected by a coefficient of we arrive at It is easy to see that this must eventually happen, since none of the functions etc., cannot have a common indeterminate divisor with the preceding one, and therefore, a division without remainder cannot happen as long as the divisor is of order greater than Thus, we will have a series of equations:


etc., up to

where etc., are integral functions of of orders where the numbers etc., continuously decrease until the last one and , etc., are integral functions of of orders , etc. (except in the case where where it is clear that we must set ).

Having prepared in this manner, we form a second series of integral functions of which we call etc., up to Indeed, let us set and for the remaining functions we derive each from the preceding two according to the same rule by which the functions , etc., are related to each other, namely through the following equations:


etc., up to

Clearly is of order here; is of order and likewise the subsequent functions , etc., are of orders , etc., so that the last one is of order

Next we consider a "third" series of functions, etc., among which any three consecutive terms will manifestly have a similar relation, namely,

Now, the first of these functions is the second is hence it is easily inferred that each is divisible by

Moreover, it follows without difficulty that we can replace the fraction with the integral function provided that no values are assigned to other than those which are roots of the equation for it is clear that the difference must vanish for such a value of since is divisible by

Instead of the function we can also take the remainder which arises upon dividing it by whose order will be lower than the order of the function

Indeed, this remainder can be immediately and more conveniently extracted using the following algorithm. We form the following equations:


etc., up to

by dividing by then the remainder of the first division by then the remainder of the second division by and so forth. Since the remainder always belongs to an order lower than the divisor, the order of the functions etc. will be respectively lower than etc.; while the last necessarily becomes since the divisor is Therefore, we have

Moreover, since only the roots of the equation are taken for it follows that etc., and under the same restriction, it follows that

However, the order of this expression will necessarily be less than than since the order of the quotients etc. must be less than etc., the order of each part etc. will be less than etc.

Finally we observe that if it so happens that among the values of the indeterminate those that need to be substituted in the fraction are a mixture of rationals and irrationals, it will be more practical to separate them and only include the latter in the equation For rational values, there will be no need for calculation; for irrational values, however, the calculation will be simpler the lower the degree of the integral function to which the fraction can be reduced.

12.

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Here is an example of the transformation explained in the preceding article. Let the given fractional function be

where indefinitely represents the roots of the equation

If we wanted to include all seven roots here, we would descend to a sixth-order integral function. However, for the rational value the calculation of the fraction is straightforward, giving the value so excluding this root from the equation of sixth degree, we have:

from which it is easily foreseen that there will arise a fourth-order integral function. Now, from the application of the preceding rules, the following sequences emerge: