William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, Second Edition, Cambridge, University Press, 1992.
Numerical computing is both the art and science.
Computers store numbers not with infinite precision but rather in some approximation that can be packed into a fixed number of bits (binary digits) or bytes (groups of 8 bits).
A number in integer representation is exact. Arithmetic between numbers in integer representation is also exact (inside the range).
In floating-point representation, a number is represented internally by a sign bit 's' (interpreted as plus or minus), an exact integer exponent 'e', and an exact positive integer mantissa 'M'. Arithmetic among numbers in floating-point representation is not exact.
The smallest (in magnitude) floating-point number which, when added to the floating-point number 1.0, produces a floating-point result different from 1.0 is termed the 'machine accuracy' epsilon_m. Pretty much any arithmetic operation among floating numbers should be thought of as introducing an additional fractional error of at least epsilon_m. This type of error is called 'roundoff error'.
for i in range(20):
print("{0} {1!r}".format(i, 1.0 + pow(10.0, -i)))
Results on Dell Lattitude with Debian 10:
0 2.0
1 1.1
2 1.01
3 1.001
4 1.0001
5 1.00001
6 1.000001
7 1.0000001
8 1.00000001
9 1.000000001
10 1.0000000001
11 1.00000000001
12 1.000000000001
13 1.0000000000001
14 1.00000000000001
15 1.000000000000001 # epsilon_m = 1e-15
16 1.0
17 1.0
18 1.0
19 1.0
Many numerical algorithms compute 'discrete' approximations to some desired 'continuous' quantity. For example, a function may be evaluated by summing a finite number of leading terms in its infinite series, rather than all infinity terms. The discrepancy between the true answer and the answer obtained in a practical calculation is called the 'truncation error'.
epx(x) = \sum_{n=0} x^n / n! = 1 + x + x^2 / 2 + x^3 / 6 + ...
sin(x) = \sum_{n=0} (-1)^n x^{2n+1} / (2n+1)! = x - x^3 / 6 + x^5 / 120 - ...
cos(x) = \sum_{n=0} (-1)^n x^{2n} / (2n)! = 1 - x^2 / 2 + x^4 / 24 - ...
Most of the time, truncation error and roundoff error do not strongly interact with one another. Sometimes, however an 'unstable' method can magnify any roundoff error and the result is useless.
https://en.wikipedia.org/wiki/Quadratic_equation
https://en.wikipedia.org/wiki/Vieta%27s_formulas
A quadratic equation is any equation that can be rearranged in standard form as a x^2 + b x + c = 0, where x is an unknown, (a, b, c) are known numbers (coefficients), a != 0. If a = 0 then the equation is linear. A quadratic equation with real or complex coefficients has two solutions, called 'roots'. These two solutions may or may not be distinct, and they may or may not be real. Discriminant: D = b^2 - 4 a c. If D > 0, then there are two distinct real roots, x1 = (-b + sqrt(D) / (2 a), # problems for b ~ sqrt(D) x2 = (-b - sqrt(D) / (2 a). # problems for (-b) ~ sqrt(D) If D = 0, then there is exactly one (double) real root, x1 = x2 = -b / (2 a). If D < 0, then there are no real roots.
Vieta's formulas [two roots are present, a != 0] a (x - x1) (x - x2) = 0, a x^2 + (-a) (x1 + x2) x + a x1 x2 = 0, x1 + x2 = -b / a, x1 x2 = c / a.