By Garrett Birkhoff
This vintage, written by way of younger teachers who grew to become giants of their box, has formed the certainty of recent algebra for generations of mathematicians and continues to be a necessary reference and textual content for self examine and faculty classes.
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Extra resources for Álgebra Moderna
In the preceding section we estimated the maximum error and the maximum relative error in the result of a calculation due to stated errors in the numbers involved. In this section we discuss the inverse problem, namely, how precise or accurate must the numbers used in a calculation be to obtain a result of preassigned precision or accuracy? We answer this question and explain the various methods by means of an example. EXAMPLE. 4: 1. 0001 sec? 5:1) I dT I ~ 21tg-11 d1T I + 1THg-! 0001. 0001), I, and g are the known quantities; I d1T I, I d/l, and I dg I are the quantities to be determined.
Aft ± b,,)(x - xo)" + ... 3:13) Co = Cl = aobl + albo , C. = aob. 3:14) f(x) = d g(x) 0 + d1(x - x ) +0 ... (x - x )" 0 + ... , where bodo + bldo bod. 3:15) If ao =F- 0, it follows from the first of these equations that division will be possible only if bo =F- 0. 2. APPROXIMATING POLYNOMIAL; APPROXIMATION AT A POINT 34 We list below some of the better known or more useful power series and their radii of convergence with some brief explanations. 1) . sm x = ~ ~ ~nB 3f + Sf ± ... + (_l)n (2n + I)!
Representation of a Function by a Polynomial. 2: I) y Let = f(x) be a function of x. 2:2) whose degree does not exceed a preassigned n and which approximates f(x) as well as possible. For the sake of brevity, a polynomial of degree not greater than n will be called a polynomial of max-degree n. " Since Pn(x) has n + I coefficients that are at our disposal, we can impose an equivalent number of conditions for the determination of the polynomial. Let us suppose first that A : (xo , Yo) is a point on the graph of y = f(x).
Álgebra Moderna by Garrett Birkhoff