C73 LAB*,' Cornell University Library QA 221.C79 A method of approximation, 3 1924 002 943 771 A ETHOD of APPROXIMATION By S. A. COREY, HiTEMAN, Iowa. [Reprinted from The American Mathematical Monthly, Vol. XIII, Nos. 6-7, June-July, 1906.] Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002943771 A METHOD OF APPROXIMATION. By S. A. COBET, Hiteman, Iowa. The follo'wiiig method of approximation may or may not be new, but as I believe it to be of practical importance I wish to call attention to it, and to point out that to secure the same degree of accuracy, it involves, at least in some cases, less labor than does the method of mechanical quadrature. It can also be used in certain cases where the method of mechanical quadrature fails, because the finite difEerences used in the latter method cannot always be found between the proper limits. On account of its rapid convergence it can also be used when other common methods fail. We have the formula,* /(^)=/(0)+i{/(.)+/(0)+2[/(-|^)+/'(Ay...+^(!?^,)]} - 5Sr[/'(^)-/'(0)]+|f5r[/''(a:) -/"(O)] -^iJ^[/'.^(^)-/"^(0)] + .... + (-^)"^^^?So!t/^"W-/^"(0)] + (1), (^1, -Ba, -B3, etc., being Bernoulli's numbers, ^, ^V; iV> ^\, A> etc.) For brevity, (1) may be written, /(x)=J'(cc,»0+^+-^+-^+ etc. ...(2), in which Sj, s^, Sg, etc., are independent of m. It is evident that m may be con- sidered a convergence factor which makes F(x, jw).approach/(«) as m approaches infinity, J" and /becoming equal as m becomes infinite. But as m increases, the labor of evaluating F{x, m) also increases, so that, in practice, it is found desir- able to obtain one or more of the higher derivatives involved in s^, s^, Sj, etc., in order to permit the use of a smaller value of m, and yet to attain the required , degree of accuracy. In certain cases it becomes exceedingly dif&cult to obtain the form and value of the higher derivatives involved in s^., s^, Sg, etc., and chiefly for this reason, it becomes desirable to eliminate from (2) certain of these quantities, (Sj, s^, Sg, etc.), after a certain degree of, approximation has been reached. To accomplish this elimination take more than one value of m and set down a separate equation for each value chosen, thus, *3ee Annals of M athe m atics , Second Series, Vol. 5, No. 4, July, 1904. Ax)=F{x, m^y m^ m*' mj (3), /(.)=J'(.,^,)+-^+^ + ^-i- (4), m m.j mw /(x)=JF(a;, >»3)+-^+Ai+A + (5). m. mi etc. (In selecting the values of m it is usually convenient to take the highest value such that one or more of the lesser values will be sub-multiples thereof.) It is evident that from (r+l) such linear equations there may be elimi- nated r of the quantities s, say s^^, Ss(i+i), ..., Sjm+r—i) • There results, /(*)= M, mT'' mj'''^'' «,r^<^+^' .. .. «ir'''+'-" M, m'' m7'''+'' mr^''+^> .. .. »»,-''*+'-'' ^(r+l) »W(r+J) mTr'^r' v-^r •• ...(6),« 1 mj^' mT'^'+'' mr*'*+*> .. .. mT'''+^'' 1 m7'^'+'^ rnV'^-'^"' .. .. m7'^'+'-'' . 1 mirfiV' .. m^r+i) • where M^, M^, ..., ilf(r+i) , represent the approximate value of /(a;) obtained by (2) for m^, m^, ..., m^r+l) , respectively, by the use of all the terms preceding Sii/m^i. If all derivatives higher than the (2i-(-2r— l)th are zero, or if all sueh even-mimlered derivatives are equal for x=x and a;=0, (6) gives the exact value of fix). In other cases the degree of accuracy attained will be no less than would have been obtained by taking the smallest value of m used in (6) and de- veloping /(a;) by (2) so far as to include all the r terms eliminated in (6). As a special case of (6), let r=2, i=2. Here M=F{x, m)-\ — ^, and we get after reducing, (i)_ j J) '{'/ Q^ m' •Determinants of this form may be readily evaluated. See Weld's Theory of Determinants , Articles 23 and 27. It is clear that the more nearly all the M's approach /(a;) , the more nearly does (6) give the exact value of /(a;), and that this increase in approximation is obtained by sufficiently increasing the integers, m, i, and r, or any of them, i The following points may be noted in passing : First. After choosing certain values of m, r, and i, f(x) is developed by (6) in linear terms of the (*•+!) quantities (ilf,, M^, ..., M^r+i) ), each ilf being an approximate value of /(a:). Second. If M—F(x, m) and if the values (but not the form) of /(a;) be given at each of the (m + l) equidistant points (0, x/m, 2x/m, ..., x) for each of the (r+1) values of m chosen, the value of /(a;) may be very closely approximat- ed, although in such case the law of its development would not be expressly stated, but it must be assumed that some such law exists in order that /(a;) may be developable in terms of x. This makes the method of practical value in many scientific problems where it becomes necessary to find /(a;) approximately, and when the only available data are a number of observations of the values of /(a;) at intervals of x/m, the approximation being closer in this method than in that of mechanical quadrature, especially where the intervals {x/m) are few. A discussion of the cases where (6) does not give the value of /(a;) is not here entered into, but the following considerations will be pertinent : First. As (6) is obtained from a number of equations, (3), (4), (5), etc., care should be taken that each of these equations actually does give a closer approximation of /(a;), when the number of terms employed include those elim- inated in deriving (6), than does M. Second. As (6), (2), and (1) are all dependent on the sum of a number of series developed by Stirling's formula, it becomes necessary that each of these underlying Stirling's series should give a true and convergent development of / between any and all of the adjacent, equidistant points (x/m) at which the value of / is taken. The following example has been chosen because its development by form- ula (1) has been given in the Monthly,* Vol. XIII, No. 4, and because its solu- tion by mechanical quadrature has been given by Dr. G. W. Hill in the Analyst, Vol. II, p. 120, 1875.t n xdx sin a;(l + .16cos^a;)2 Taking r=2, i=2, m,=6, m2=3, m^=2, we get Jfj =1.657,626,355, Jlfj =1.657,490,406, ^£3=1. 657,013,853. Substituting in (7), we get, *The value of the integral is incorrectly given in the Monthly as, 1.657,636,524, in- stead of 1.657,636,257. The error is due to the fact that r 2596.^ -1 ■' ' L4200.128.6!j is carried out as .000,000,009 instead of .000,000,276. tSee also Hill's Collected Works, Vol. I, p. 204. f,r. _f(ns^ C^' ^ 1728ilf, - 23328Jlfa + 233280Jlf ^ ■^ ^ A ; J^ sina:(l + .16cos2a;)? ~~ 211,680 =1.657,636,33. ..(8). We may check the accuracy of the work of computation and determine the degree of approximation attained in (8) by finding the values of /(-q-) and f (-g-)- This done we may take m,^—6, wig— -4, ni^=Z. Substituting these values of m^, m^, m^, in (7) we get an expression similar to (8) which gives a value of the definite integral coinciding with the value found in (8) for six deci- mal places, thus proving the accuracy of the work and showing the number of decimal places to which result found by (8) is correct. . This result is not as accurate as that obtained in the April Monthly for the reason that the smallest value of m here used is 2. This is therefore a more accurate result than would have been obtained by taking m=2 and developing by (1) so far as to include the term involving B^, but a less accurate result than was obtained in the Monthly by taking m==6 and developing by (1) far enough to include the term involving B^ . The above result is, however, as accurate as Dr. Hill's and was obtained with much less labor. To obtain a more accurate result than that given in the Monthly without finding any higher derivatives than are there given, take i=4, r=l, mj=6, »M2=3, and substitute in (6). By using nine decimals throughout the result is found to be 1.657,636,259. By using ten deci- mals throughout a result correct to nine or ten decimal places would have been obtained. On a Set of Four Linear Associative Alge- braic Units. By S. A. COREY, Hiteman^ Iowa. [Reprinted from The American Mathetnatical Monthly, Vol. IV, No. 2.] THE AMEKICM MATHEMATICAL MONTHLY. Entered at the Post-ofRce at Springfield, Missouri, as second-claas matter. VOL. XIV. FEBRUARY, 1907. NO. 2. ON A SET OF FOUR LINEAR ASSOCIATIVE ALGEBRAIC UNITS. By S. A. COREY, Hiteman, Iowa. Dr. Dickson has called attention in the November Monthly to a very, remarkable set of non-associative algebraic units, and notes that among the many other similat- (though usually associative) units, the best known are Hamilton's famous unit vectors, i, j, k, which on account of being perfectly isotropic when applied to Euclidean space, have been of much use to scien- tists in simplifying the language of analys is as applied to the physical sciences. In this connection it may be of inferest to notice very briefly another somewhat remarkable set of "units, "in terms of which, when inter- preted in a certain manner, scalars, vectors, and quaternions may all alike be written. We shall begin, as we may, by defining these "units" by a "multiph- cation table." This is virtually what Hamilton did when he defined his "units" in such a way as to admit of their bearing the interpretation he desired to place on them as unit vectors. Let, then, H, I, J, K, be these four "units" of which the following is the "multiphcation table:" H I J K H H I H -I I H I -H I J J- -K J K K -J K J K Further assume that multiplication of these "units" with ±1 is always commutative, e. g. H{—J) = — HJ. Then will multiplication always be associative; for, taking these "units" in sets of three, thus, H.IJ^-HH=-H, HLJ=IJ=^-H, J.HK=-JI=--K, JH.K^JK=K, and so on until all possible permutations of the four "units" taken three at a time have been enumerated. In every case multiplication will be found to be associative. Similarly taking the four "units" four at a time, thus, HIJK^H. IJK=HIJ. K=H. I. JK==H. IJ. K=HI. JK=IK=I, HJIK^H.JIK=HJI. K=H. J. IK=H. JI. K=HJ. IK^HI=I, and so on, until all possible permutations of the four "units" taken four at a time have been enumerated. In every case multiplication will be found to be associative. But as any and all permutations or combinations of the four "units" taken five or more at a time can readily be reduced to some of the foregoing permutations of four letters, it follows that multiplication is always associative. After thus defining our "units" we may impose further restrictions, provided only these restrictions in no way conflict with the restrictions im- posed by our "multiplication table." By so doing we simply exclude from consideration all interpretations of these "units" which do not admit of these added restrictions. Let these added restrictions be as follows: First. Our "units" may be added (and subtracted) . Second. Addition (and subtraction) must be associative, { [i7 + 7) +J -VK'\=iH+ U+J) +K], and so on}. Third. Addition (and subtraction) must be commutative, [(H+I+J +K) = {J+H+K-^I), and so on]. Fourth. Multiplication must be distributive over addition, [H{H+I +K+J) = iHH+HI+HK+HJ), and soon].' Fifth. Multiplication, addition (and subtraction) of our "units" with ordinary scalars (rational and irrational) must be associative, commutative, and distributive, exactly as if these ' 'units' ' were ordinary scalars. It should be particularly noticed that no assumption has been made that division is always possible, nor that the law of indices holds. For this reason no operation on our "units" involving either division or the law of indices is permissible, until, in any particular case, these operations have been proven permissible. We may now seek to interpret the meaning of these "units" in such a way as to satisfy all the imposed conditions or restrictions. With this pur- pose in view it may be noted that all the restrictions imposed, except those imposed by the "multiplication table" are the same as those that apply to algebraic numbers (scalars) and vectors. We know that vector multiplica- tion is non-commutative. In view of these facts it may be worth while to try to so combine algebraic numbers and vectors as to form combinations the multiplication of which is in accordance with the above "multiplication table." For, should we succeed in forming such combinations, they must fulfill all the requirements imposed on our "units," and, therefore, may be treated as particular values of these ' 'units. ' ' No great difiiculty is involved in finding a number of combinations which do not change in value when squared (none of our "units" change in value when squared), and but little ingenuity is required to combine a set of four of them in such a way as to fulfill all the stated requirements. One such set is H=i[l-j-e{i+k)] i=--m+j+Hi-m J=m+3-Hi-k)'\ K=i[l-j+o{i+k)-\ whence l=hiH+I+J+K) , ^^^ i=-^iH-I+J-K) j=ii-H+I+J-K) k=^{H+I-J-K) where ^=i/— 1, and i, j, ank k are Hamilton's unit vectors. Other similar combinations can readily be obtained from this set by a cyclical permutation of the unit vectors, i, j, k, or by substituting for these vectors other rectan- gular unit vectors. All such combinations must, however, be considered as mere variations of {A). We know that in ordinary algebra we may substi- tute such operators as d/dx, d/dy, d/dz, etc., for ordinary algebraic num- bers because their laws of combination are identical. Whether, likewise, such distinct values of H, I, J, and K exist is a matter of speculation, but seems not impossible. It might have been a more logical and simple process to have com- menced by assuming (A) and then deducing the "multiplication table" there- from. Such a method of procedure would have led up to the same results, but would not have made it so evident that the values of the fundamental "units" given in (A) are mere particular values. By closely examining (A) we learn that this interpretation of our "units" impHes the following: First. Each "unit" is a Cayleyan nullitat (quaternion with zero norm) . Second. All quaternions, including scalars and vectors, can be writ- ten in terms of H, I, J, and K, and vice versa. Third. These "units" maybe algebraically combined in accordance with the same laws that govern the combination of quaternions. (If divis- ion and the law of indices be excepted, there seems to be no reason why the laws of quaternions should not govern in the combination of other entirely distinct interpretations of these "units," should such interpretations be found. ) Let Q=iaH+bI+cJ-{-dK), R^ieH+fl+gJ+hK), W^irH+sI-\-tJ +uK), a, b, c, d, e,f, g, h, r, s, t, and u, being numbers in the algebraic field, operators, such as d/dx, d/dy, d/dz, etc., or a combination of such numbers and operators. Then, if Q.R=W, r^e{a+c)+f{a-c), s=e{h-d)+f{h+d), t=g{c + a)+h{c—a), and u—g{d—h)+h{d^h). Using ordinary quaternion methods and Hamilton's notation, SQ=h{a+h+c+d), VQ^iQ-SQ) = ia-w)H+{b-w)I+ic-w)J+id-w)K, [w^iia+b+c+d)], TQ=]/[2{ad+bc)], and UQ-^i2ad ^r2bc)-HaHHI+cJ+dK). Similar expressions can readily be found for SUQ, VUQ, TVQ, TVUQ, etc. It would, indeed, be possible to construct an entire system of quaternions in which H, I, J, K would be the "units" used to replace unity and the i, j, k of Hamilton. Such a system would necessitate the use of irrational numbers in expressions involving ordinary vectors, and would, therefore, not be well suited for use in problems which concern the physi- cist; it would, however, have the remarkable characteristic that scalars, vec- tors, and vector products could all alike be expressed in terms of the same linear, homogeneous units, although such expressions would be cumbersome and of little practical value to the physicist. It is a matter of historic inter- est that Hamilton was much concerned about the heterogeneous character of his vector products, and tried to find an "extra-spacial" unit which would render such products homogeneous. * The ]'—l of algebra may, perhaps, not be considered such an "extra-spacial" unit, but may evidently be used to obtain the result Hamilton sought to obtain by the use of such a unit. Inasmuch as the heterogeneity of vector multiplication disappears when |/ — 1 is introduced by the linear substitutions involved in these "units," we may, perhaps, better conclude that the heterogeneity which Hamilton sought to remove was, after all, only a seeming heterogeneity resulting from the point of view afforded by his particular system of fundamental unit vectors, whereas quaternion analysis as a system is, of course, independent of any particular set of fundamental unit vectors. But, in conclusion, we may observe that the algebraic properties of these "units" had been defined and proven consistent before any attempt was made to give them an interpretation. Had we, indeed, entirely failed to find an interpretation, these "units" would, nevertheless, have been real- ities, in an abstract sense, to the pure algebraist. The fact that they hap- pen to be isomorphic to a certain set of four imaginary quaternions is, of course, not without interest to him, but should be looked upon by him as a mere coincidence. *See article on Quaternions by Professor Tait in Encyclopedia Brittannica. Makers Syracuse, N. Y P«I. m 21, 1908