500. Contraction of Horner's Method. In § 498 the student
will see that if we seek only the first six figures of the root,
the last six figures of the fourth coefficient of the last trans-
formed equation may be rejected without affecting the
result. Those figures of the second and third coefficients
which enter into the fourth coefficient only in the rejected
figures may also be rejected. Moreover, we may reject all
the figures which stand in vertical lines over the figures
already rejected.

The work may now be conducted as follows:

The double lines in the first column indicate that beyond this stage of the work the first column disappears altogether.

In the present example we find three figures of the root. before we begin to contract. We then contract the work as follows:

Instead of adding ciphers to the coefficients of the transformed equation, we leave the last term as it is; from the last coefficient but one we strike off the last figure; from the last coefficient but two we strike off the last two figures; and so on. In each case we take for the remainder the nearest integer. Thus, in the first column of the preceding example we strike off from 174 the last two figures, and take for the remainder 2 instead of 1.

The contracted process soon reduces to simple division. Thus, in the last example, the last two figures of the root were found by simply dividing 897 by 800.

To insure accuracy in the last figure, the last divisor must consist of at least two figures. Consider the trial divisor at any stage of the work. If we begin to contract, we strike off one figure from the trial divisor before finding the next figure of the root. Since the last divisor is to consist of two figures, the contracted process will give us two less figures than there are figures in the trial divisor.

Thus, in 498, if we begin to contract at the third trial divisor, - 79,908, we can obtain three more figures of the root; if we begin to contract at the fourth trial divisor, - 800,213, we can obtain four more figures of the root; and so on.

The student should carefully compare the contracted process on page 486 with the uncontracted on page 484.

501. When the root sought is a large number, we cannot find the successive figures of its integral portion by dividing the absolute term by the preceding coefficient, because

the neglect of the higher powers, which are in this case large numbers, leads to serious error.

Let it be required to find one root of

x1-3x2 + 11 x - 4,842,624,131 = 0.

(1)

By trial, we find that a root lies between 200 and 300. Diminishing the roots of (1) by 200, we have

y* + 800 y3 + 239,997 y2+ 31,998,811 y 3,242,741,931 = 0. (2)

The signs of f(y) show that a root lies between 60 and 70. Diminishing the roots of (2) by 60, we obtain

z1 + 1040 z3 +405,597 z2 + 70,302,451 z — 273,064,071 = 0.

(3)

The root of this equation is found by trial to lie between 3 and 4. Diminishing the roots by 3, we may find the remaining figures of the root by the usual process.

502. Any root of a number can be extracted by Horner's Method.

If the number be a perfect power, the root will be obtained exactly.

503. From the preceding sections we obtain the following general directions for solving a numerical equation :

I. Find and remove commensurable roots by §§ 491–493, if there are any such roots in the equation.

II. Determine the situation, and then the first figure, of each of the incommensurable roots as in § 494.

III. Calculate the incommensurable roots by Horner's Method.

Exercise 143.

Calculate to six places of decimals the positive roots of the following equations:

1. x3-3x-1= 0.

2. x+2x-4x-43 = 0.

3. 3x3 +3x2+8x320.

4. 2x3-- 26x2 + 131x - 202 = 0.

5. x12x+7=0.

6. x-5x+2x2 - 13x+55 = 0.

Calculate to six places of decimals the real roots of the following equations, when incommensurable:

504. The problem of determining the number and situation of the real roots of an equation is completely solved by Sturm's Theorem. In theory Sturm's method is perfect; in practice its application is long and tedious. For this reason, the situation of the roots is in general more easily determined by the methods already given.

Before passing on to Sturm's Theorem itself we shall prove two preliminary theorems.

505. Sign of f'(x). The function f'(x) is by definition (§ 476) the limit of the ratio of the increment of f(x) to the corresponding increment of x, as the latter approaches 0 as a limit.

If we suppose the increment of x to be always positive, then the corresponding increment of f(x) may be positive or negative. If the increment of f(x) is positive, however small the increment of x may be, then in the limit f'(x) will be positive. But if for very small increments of x, the increment of f(x) is always negative, then f'(x) will be negative.

In other words, f'(x) is positive or negative for any value of according as the function f(x) is increasing or decreasing as x increases from this particular value of x.

Referring to the graph of ƒ(x), ƒ'(x) will be positive as long as the curve is rising toward the right, and negative

O at these points. The values of x corresponding to A and B are roots of the equation ƒ(x) = 0.

506. Signs of f(x) and f'(x). Let a be any real root of an equation f(x) = 0, which has no equal roots.

Let x change continuously from a -h, a value a little less than α, to a +h, a value a little greater than a. Then f(x) and f'(x) will have unlike signs immediately before x passes through the root, and like signs immediately after x passes through the root.

For, in the graph, if f(x) is positive just before x passes through a root as at P and R, the curve is sinking to the right, and therefore f(x) is negative, both before and after