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5. Given X = x3 + px + q = 0, to find the conditions which will render all the roots real.
6. For the same purpose, given x2 + px2 + qx + r = 0.
Let the given equation be Ax" + Bæ”¬1 + Cœ”−2 + Lx2+ Mx+N=0; and let a be a distinct approximation* to one of its roots; it is required to evolve the remaining figures in succession.
1. Reduce the roots of the equation by a,, and denote the reduced equa
If, however, upon transforming, the sign of N, should prove to be different from that of N1, whilst that of M, is the same with that of M1, the value a, is too great, and the next smaller number of the same decimal denomination should be tried, till one is found which fulfils the requisite condition.
=a, find a new root-figure, and transform as before and proceed thus till all the figures are found, if the root terminate, or as many as may be necessary if the root be interminable.
4. When the root is negative, change the signs of all the roots (Prob. IV.)
* By a distinct approximation is meant a value a, which is nearer to r1, a root of ƒ(x) = 0, than the corresponding root p1, of fi(x)=0 is to r1. In this case ƒ(a) and ƒ¡(a) have contrary signs; and for the most part (almost always after transformation by the first decimal of the root) the quotient —ƒ(a)÷fi(a) gives the next figure of the root accurately, and the more especially if none of the intermediate coefficients are comparatively large numbers. This is obvious from Theor. XIII. as then one of the roots of the reduced equation is relatively very small; and as we proceed to diminish the roots still further, by succeeding decimals of the root to which we are approximating, it becomes accurate to several decimal places..
and find the positive roots above. These written negatively, will be the negative roots of the equation.
5. In actually working out the transformations, it will be convenient to mark the resulting coefficients of the transformed equation by oblique lines, as in the example on p. 234, instead of recommencing the work by writing them anew in a horizontal line.
6. After obtaining two or three decimals of the root, the work may be very much contracted, analogously to that employed in contracted multiplication and division of decimals, in the following manner:
(a). Let A, x," + B, x,”1 + .... + L, x,2 + M, x, + N, 0 be the reduced equation after which the contractions are to commence: then draw a vertical line on the right of the figures in N,; a vertical line cutting off one figure from M,; a vertical line cutting off two figures from L,; and so on, till 1 figures are cut off from B,, and n figures from A,.
(b). Find with this contraction the next figure of the root ap+; and reduce by this, taking the figures to the left of the vertical lines, with one of those on their right as the multiplicand in each case, (taking care, however, to estimate the effect for the purpose of " carrying" of the rejected ones, as near as possible,) and put the results down in the corresponding places of the next column; viz. beginning with that on the right of the vertical line. The additions to be performed as when there was no contraction of the work. This will give Ap+1 xp+1
Mp+1 Xp++ N, = 0.
(c). In the next transformation, cut off one, two, three, (n - 1) figures from the contracted coefficients, Mp+1, Lp+ + Kpt.... B+; and proceed as before.
In these processes, as the greatest number of figures is cut off from those columns which originally contained the fewest, these will diminish very rapidly; and after a few transformations, an equation of a high degree is reduced in point of simplicity to one of a low degree; and generally the last half of the entire figures of the root are obtained by contracted division only.
Moreover, if p figures have been found, and ʼn be the degree of the equation, the number of figures of the root which may be trusted to as quite accurate, will be np1. Thus in an equation of the 5th degree, if three decimals have been found, the contraction will give 3 . 5 114 places true in all.
The theory of this contraction is very simple, but it does not admit of being concisely laid down in words: but a little consideration will enable the student to perceive that the effect of the parts cut off falls entirely to the right, in all the columns of the correction column, or of that which follows the column of figures to the right of the vertical lines.
- 2x3 +10x2
In illustration of the entire process, let the equation x + 4x1· 9620 be proposed, in which a distinct approximation to one of the roots is 3. Then, performing the reduction by 3, we have
To find a,, we have a2 = —
= 4; but upon reducing the equation
by. 4, we find N2 = + 40·68064, or the sign of the final term is changed. Whence try a2 = 3, and the condition is fulfilled. In all the subsequent stages
- N2 Mp
= ap condition. The following is the process.
is found upon performing the reductions, to fulfil the required
This result is rather too small, owing to the contracted corrections of the coefficients, but especially of the fourth, being kept uniformly above the truth: but they have been retained to show the manner of conducting the operation, instead of throwing out the zeros which would have taken the place of the units now to the left of the line.
GENERAL RECAPITULATION AND REMARKS.
1. Count the number of variations and the number of permanencies of sign in the given equation; there will be as many positive roots as variations of sign, and as many negative roots as permanencies of sign. (Theor. VIII.)
2. If there be an odd number of positive roots, one at least of them will be real; and if an odd number of negative roots, one at least of these will be real.
3. As an initial experiment, reduce the roots of the equation, both as it is given, and with its alternate signs changed by the factors of the absolute term; since, if there be any integer roots, they are factors of that term, and in such case, the first horizontal line of operations will render N1 = 0. If any such be
found, then employ in like manner the factors of M, of the reduced equation; and so on, as long as the division terminates. The integer roots will all be thus easily found. See Example 4, p. 236.
4. No equation having all its coefficients integers, and A different from unity, can have a fractional root: all such equations must, therefore, have their real roots either integers or interminable decimals.
5. If one variation be lost in passing from a transformation from an integer a to its consecutive integer a + 1, then the number represented by a is the principal part, or "first figure" of the root.
6. If two, four, or any even number of variations be lost in the transition, there are two, four, or some corresponding even number of roots, in the interval, of which one, two, or some corresponding number of pairs may be imaginary, and the remainder real.
7. In this case consider which criteria are most likely to be applicable to the determination of the number of real roots.
If De Gua's apply, it will be the most simple; or if any inference can be drawn from it under the aspect presented in the note on p. 225, let it be done.
In case of still doubting the character of the roots, apply Budan's criterion first of all, as directed in the statement, p. 226, 1 being the reciprocal of 1. If there be still uncertain roots, (or m less than n,) proceed as in the foot-notes, using either the reduced reciprocal equation, or a narrower interval in the original direct one.
Should there still be any doubt, which can never be the case except there be equal roots, or roots having very minute differences, have recourse to Sturm's Criterion. This in its progress, by giving some one of the modified remainders X, = 0, will furnish the component equation containing the equal roots; and if there be not equal roots, the functions so derived will furnish a complete criterion for every part of the series of values between r, the greatest, and r,, the least, of the roots *.
8. When there are two real roots in a small interval, it will always be more convenient to seek one or two figures of the corresponding roots of its reciprocal equation in the outset, as suppose a and b : then the leading parts of the roots
of the given equation will be found from and, and the approximation continued as usual from these.
9. When the root of an equation has been accurately determined, use the depressed equation for finding the other roots: but when a root has been ap
* This may be in some degree an apparent inversion of the natural order of proceeding to obtain a complete solution of the equation. The most obvious course would be: (1), to form Sturm's functions X, X1, X2, ... X», which in their progress would detect the equal roots (2), to apply the limiting integers a and a + 1 to these functions in all such cases as presented a doubt: (3), to employ narrower intervals for finding the distinct approximation: and (4), to develop the roots whose initial values had been found by Horner's method. However, the very great labour attendant on finding Sturm's functions, renders it desirable to evade their use if it can possibly be dispensed with; and this can almost always be done by taking narrower limits for the transformation, as we thereby, for the most part, separate the pairs of roots which occur at small distances in the numerical scale from each other; and there is never any difficulty in determining by Budan's criterion, as modified in the notes, whether these be real or not. The order of working, therefore, pointed out in the text, contributes greatly to expedition, whilst it is much less likely to be productive of numerical error than the complicated operations and unwieldy numbers that are essential to Sturm's operation.
proximately determined, return to the original one (or an accurately depressed equation, if such has been found, by means of an accurate root) to find the other roots.
10. Equations whose coefficients are rational, cannot have irrational or imaginary equal roots, without their conjugates; or, in other words, if there be equal roots of the form a + b√ ± 1, there will also be as many of the form a-b± 1.
Ans. 1, 1, 235, and the roots of x2+237 +55698 = 0. 2. Find the roots of x6 + 4x5 8x4 25x3 + 35x2 + 21x ·
28 = 0. Ans. - 4, - 1, 1, 1·356896, 1.692021, and 3.048917. 1010, and of x2 10x100, by the general
3. Find the roots of x2
100498756 in the former, and 16 18034, and
6.18034 in the latter.
4. Solve x-6x7— 12x + 134x3 — 289x1 + 480x3— 660x2 — 608x + 960 = 0.
Ans. 4.54419552, and four imaginary roots.
6. Determine completely the characters of the roots, and the values of the real ones in the following equations :
30; (all real.)
302x + 2000; (two imaginary.) 350 = 0; (two imaginary.)
7. Find what roots are imaginary, and find the real ones, in
8. To find the values of x and y, there are given the two following:
9. Given xy2+ x2y + 1·75 = 0, and x2 + y2 = 4·25, to find x and y. 10. Given x2 + yz = 16, y2 + zx = 17, and z2 + xy=18, to find x, y, z.
In all the inquiries which have preceded the present, in this course, we have been required to find the special values of one or more unknown quantities, so as to satisfy the given equations, or the conditions which they expressed. There is, however, a distinct class of inquiries, in which we are required to change the form of any given compound expression into a series of single terms. If the indicated operation be one which we know how to perform, this change may in general be effected by an actual performance of those operations: but it will