STURM'S THEOREM. 



STURM'S THEOREM. 



$60 



The came oonchuions will be found from other cases, and we hare 

 mow examined every way in which the criterion can undergo an altera- 

 tion in the order of the sign* of which it is composed. And since, the 

 function K'ing of dimensions, there are altogether changes, and n 

 only, to low, it follows that every pair of signs lout by the vanishing 

 of any of the derived functions, in any internal part of the criterion, 

 how that there must be two imaginary roots : for there must be N 

 roots only, every real root must be accompanied by a change lost at 

 the head of the criterion, and every loss of changes which takes place 

 anywhere else diminishes the number which can take place at the 

 head. Again, since Kisses other than at the head of the criterion must 

 take place in even numbers, it follows that of any odd number of 

 losses, one must have been effected at the head, or must have risen 

 from a real root ; or if not one, some other odd number. 



The manner in which the changes) of sign take place is as follows : 

 When f "<*>, or even when it is numerically greater than any 

 negative root, the criterion presents nothing but changes. Alterations 

 of the criterion consist in : 1, Loss of one or more changes at the head 

 of the criterion (showing real roots); 2, loss of changes in even 

 numbers in the middle of the criterion (showing imaginary roots) ; 8, 

 deration of changes, or alteration of their place in a direction towards 

 the head of the criterion. This last takes place only when an odd 

 number of derived functions vanishes, the including functions (pre- 

 ceding and following) having different signs. So soon aa a root has 

 been pasted, there is a permanency + + or -- at the head of the 

 criterion; before another root is arrived at, this permanence niuat 

 have become a change, since a change there must then be at the 

 head to be lost in passing through the root Hence it follows that 

 between two roots of ip.c U, there must lie a root of <f'.c = 0; and 

 this root is either single, triple, quintuple, &c., but not double, 

 quadruple, Ao> 



For example, let <)>x=x t -~x t +lSx i lO.r + 2, 

 * 10, ^a^e* 3 21.T+15, 



There are no negative roots, as is obvious from their being nothing 

 but changes among the co-efficients ; if we construct the criteria for 

 i= 0, 1, 2, 3, and 4, we find the following results : 



<t>x 



1 



+ 

 + 







2 3 



i. 



4 



+ 



t- 



+ 



When z=Q, the criterion shows four changes; at .r=l, it is inde- 

 finite, owing to .1 = 0. But immediately before a: = 1, ^ 2 .r must, by 

 the preliminary theorem, have the sign contrary to that of Q t x, or the 

 rign +; consequently, for x=\ h, however small A may be, the 

 criterion must be + + + (- . Two changes of sign are therefore 

 lost in passing from x=0 to x=l, and there are either two real roots 

 between and 1, or two imaginary roots. To try this further, let 

 *= J, the criterion of which is + + + ; so that there is one root 

 between and J, and another between ^ and 1. When :r=] + A, the 



criterion is + H h, so that there is no root between 1 and 



2. Lastly, there is one root between 2 and 3, and one between 3 

 and 4. 



The theorem of Fourier, though very convenient in practice, is 

 defective in theory, as requiring an indefinite number of trials. If two 

 roots were very nearly equal, it would require very minute subdivision 

 of the interval in which they are first found to lie, to distinguish them 

 from a pair of imaginary roots. This theorem waa not published till 

 1831, in Fourier's posthumous work, but its author had made his 

 methods known, and among others to the late M. Sturm, a young 

 Qenevese, employed in the bureau of M. de Ferussac, editor of the 

 bulletin which bore .his name, afterwards a member of the Institute, 

 who died a few years ago. Sturm applied himself to the detection of 

 functions which should stand in the place of (ps, <t> t x, <t> t x, &c., in such 

 manner that the criterion formed from them, in the same way as in 

 Fourier's theorem, should never lose a change of signs except in pass- 

 ing through a real root. In this he signally succeeded ; and thus, 

 though his theorem presents great practical prolixity of detail, he fur- 

 nished a complete solution of the difficulty which hod occupied 

 analysts since the time of Descartes. This theorem may be proved 

 as follows : 



Let there be any number of functions v, v,, v,, . . . . v r , the last of 

 which is a constant independent of x, and all but ;thc last are functions 

 of .r. Let them be connected together by the equntii >n 



p,p,, Ac., being any functions of x, which do not become infinite 

 when v,v,, Ac., vanish. From this it follows, first, that no two con- 

 secutive functions of the set v, v,,Ac., can vanish together; for if 



v, And v,, for instance, vanished together, the third equation shows 

 t li.it v, would also vanish, the fourth that v, would vanish, and so on ; 

 consequently v r , a given constant, also vanishes, which is alumni. 

 Secondly, when any one after T vanishes, the preceding and following 

 must have different signs; fory,=0 gives v= v,, v, = gives v,= 

 v,, Ac. Now call the signs of v, v,, Ac., the criterion, and let v = 

 when a-n, there being only one root of that value, so that v changes 

 sign in pawing from *=a A to x = a + h. Since v, does not vanish 

 with T, we have one of the four cases following : 



V 



V, 



V 



v. 















If v, lie the derived function of v, only the second or third cioai 

 can happen, by the theorem so often used in the preceding part of this 

 article; so that a change of sign will be lost at the head of 

 for every single root of v = 0. Nor will any change of sign <-\ 

 gained or lost in any other manner; for suppose .i- = o give* v, = 

 for instance, then v, and v, have different signs, and in passing from 

 #= A to =>a + A, if each be so small that no root of v, or v s 

 lies between a + A and a -A, we must have one of the following 

 cases : 







+ 



In no one of these is any change of sign lost, or anything except a 

 change and a permanence when js= a A, and a change and a perma- 

 nence, in a different order perhaps, when .r=a + h. Consequently, if in 

 passing from x=a, the less, to x =b, the greater.it appear that no 

 changes of sign are lost, it is certain that there must have been no real 

 roots of v = between x = a and x =4. 



Now, v, being the derived function of v, it remains to find v.,, v,, 

 Ac. Divide v by v,, which is of one dimension lower, and we have a 

 quotient, say P 1( and a remainder B,. Then v = p, v + n or v.,= n. 

 Again, divide v, by V a , giving a quotient P s , and a remainder it., : we 

 have then v, = p, v, + n a or v s = R^i a d so on. It appears, then, 

 that v r> being the derived function of v, we must proceed as in finding 

 the greatest common measure of v and v,, only changing the sign ot' 

 every remainder as fast as it is obtained. In order that the last, v r , 

 may be a finite constant, it is requisite that there should be no equal 

 roots. We must then suppose the equal roots to be separated before- 

 hand, as in the usual method. In fact, this very process of finding the 

 greatest common measure, with 6Y without change of sign in the 

 remainders, will first detect the equal roots, if any. It is important 

 to remark, that at any step multiplicatiou by any positive quantity in 

 allowable, the signs (the only things we have to do with) not being in 

 any case altered by such multiplication. 



In INVOLUTION AND EVOLUTION a method of performing the 

 operations required in Sturm's theorem was proposed, which avoids 

 useless writing. Mr. Young (' Math. Dissertations,' p. 143) pro- 

 posed another, of much the same degree of abbreviation. Sturm's 

 theorem however requires so much operation, that there can be little 

 doubt of that of t ourier being a more easy mode of workiug any 

 particular case. It is not however as a key to the mere numerical 

 solution of equations that either of these theorems must be viewed : 

 the insight which they give into the nature of equations, and still 

 more that which they are likely hereafter to give (for neither is more 

 than a germ), will render them both important steps in the progress of 

 algebra. 



Since it is sufficient to the theorem that the last function \, should 

 retain one sign, and not vanish, we may stop in the process when 

 we arrive at any function of which all the roots are known, or can 

 be discovered, to be impossible. And it is easily shown, as was done 

 by Sturm himself, that even when there are equal roots, so that the 

 last, v r , is neither constant nor always of the same sign, the theorem 

 still remains true, so far as to give the number ot diflrtnt roots 

 which lie between any two given limits, without any information as 

 to the number of times which each root should be repeated. 



For instance, in the article cited we find 



v = *-a* 1 



We need not go further, for v, has none but imaginary roots. Now 

 when a;= oo , the criterion is -i --- I- ; when * = 0, it is H + : and 

 when .1= + oo , + + + . Consequently there is one negative root, one 

 positive root, and a pair of imaginary roots. 



The following example is from Mr. Young (p. 191) : it is an instance 

 riven by Fourier in illustration of his own method, and Sturm's is 

 applied to it by Mr. Young, to show (the superior certainty of the 

 latter. Of that certainty no one can doubt, but the process exhibited 



