math.md

Math functions

Mathematical functions are essential in all computing environments. R also provides the basic math functions.

Trigonometric functions

Symbol	Call	Value
$$\sin(x)$$	sin(1)	0.841471
$$\cos(x)$$	cos(1)	0.5403023
$$\tan(x)$$	tan(1)	1.5574077
$$\arcsin(x)$$	asin(1)	1.5707963
$$\arccos(x)$$	acos(1)	0
$$\arctan(x)$$	atan(1)	0.7853982

Hyperbolic functions

Symbol	Call	Value
$$\sinh(x)$$	sinh(1)	1.1752012
$$\cosh(x)$$	cosh(1)	1.5430806
$$\tanh(x)$$	tanh(1)	0.7615942
$$\mbox{arcsinh}(x)$$	asinh(1)	0.8813736
$$\mbox{arccosh}(x)$$	acosh(1)	0
$$\mbox{arctanh}(x)$$	atanh(1)	∞

Number functions

Symbol	Call	Value
$$x!$$	factorial(5)	120
$$\lceil x\rceil$$	ceiling(10.6)	11
$$\lfloor x\rfloor$$	floor(9.5)	9
truncate	trunc(1.5)	1
round	round(pi,3)	3.142
significant numbers	signif(pi,3)	3.14

Extrema functions

Symbol	Call	Value
$$\max(\ldots)$$	max(1,2,3)	3
$$\min(\ldots)$$	min(1,2,3)	1

Root finding

polyroot() can find complex roots of a polynomial equation in the form of

$$ p(x) = z_1 + z_2 x + \ldots + z_n x^{n-1}.$$

For example, find roots of the following equation:

$$ 1 + 2 x + x^2 - x^3 = 0 $$

> polyroot(c(1,2,1,-1))

[1] -0.5739495+0.3689894i -0.5739495-0.3689894i  2.1478990-0.0000000i

Note that all complex roots are found.

As for general numeric root finding in the form $$f(x)=0$$, uniroot() function can be used to numerically find a root of that equation.

For example, find a root of the equation $$x^3 - x + \cos(x) = 0$$ within the range $$x\in[-5,5]$$.

> uniroot(function(x) x^3-x+cos(x),c(-5,5))

$root
[1] -1.159601

$f.root
[1] 1.864754e-05

$iter
[1] 11

$init.it
[1] NA

$estim.prec
[1] 6.103516e-05

In the function call, we pass an anonymous function to uniroot(). We will cover this in detail in later chapters.

Calculus

It is very handy to perform basic calculus.

Derivatives

D() computes the derivative of a function symbolically with respect to given variables.

For example, derive $$\mbox{d}x^{2}/\mbox{d}x$$.

> D(expression(x^2),"x")

2 * x

Derive $$\mbox{d}\sin(x)\cos(xy)/\mbox{d}x$$.

> D(expression(sin(x)*cos(x*y)),"x")

cos(x) * cos(x * y) - sin(x) * (sin(x * y) * y)

Thanks to the expression() function that keeps the expression unevaluated so that the symbols are directly accessible. Expression object gives R the power of meta-programming. We will cover this topic in advanced chapters.

Since the derivative is also an unevaluated expression, we can evaluate it given all necessary symbols by calling eval().

> z <- D(expression(sin(x)*cos(x*y)),"x")
> eval(z,list(x=1,y=2))

[1] -1.75514

Integration

R also supports numeric integration. Here we do not have to write the expression but provide a function since it is not symbolic computation. For example, the following code calculates

$$\int_{0}^{\frac{\pi}{2}}\sin(x),\mbox{d}x.$$

> integrate(function(x) sin(x),0,pi/2)

1 with absolute error < 1.1e-14

Since its numerical computation, it inherits all the pros and cons of such computing technique.