# Numerical differentiation in R

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## Outline

• The following topics will be covered in this lecture:
• Concepts in analytical differentiation
• An approach to numerical differentiation
• Newton’s method in one variable.
• Functions of multiple variables.
• Multivariate Taylor approximation.
• Jacobians
• Newton’s method in multiple variables

## Concepts in analytical differentiation

• Using the rules of calculus, e.g.,

1. the product rule;
2. the power rule;
3. the chain rule;
• we can compute derivatives of complex functions analytically.

• However, if the function is only given in an implicit form, e.g., as the output of an algorithm, we still need approximate ways to compute such derivatives.

• We will begin by introducing concepts in analytical derivation and then discuss one approach to their approximation.

• This will include some examples on how to compute such analytical derivatives and approximations in R.
• We will follow this with one of the primary uses of such an approach, solving systems of nonlinear equations.

### Concepts in analytical differentiation

• Recall, R has the ability to recognize mathematical expressions as objects.

• E.g., let us create the following expression:
f <- expression(x^3 * cos(x))

• R also has a means to differentiate such expressions automatically;

• This is with the function 'D' that takes a syntax of the form
D(expression, variable_name)

• Q: suppose we use f as the expression and x as the variable name, what will be the result?

• A: we can compute the above derivative analytically using a combination of the product and power rules to obtain:
D(f, "x")

3 * x^2 * cos(x) - x^3 * sin(x)


### Concepts in analytical differentiation Courtesy of Pbroks13, CC BY-SA 3.0, via Wikimedia Commons

• The derivative represents the slope of a tangent line to a curve.
• In the figure to the left, we see the function $$f$$ represented by the blue curve.
• The derivative $$f'(x)$$ at a given point gives the infinitesimal rate of change at that point with respect to small changes in $$x$$, denoted $$\delta_x$$.
• Suppose we have a point $$x_0$$, a nearby point that differs by only a small amount in $$x$$ $x_1 = x_0+\delta_{x_1},$
• The function $f(x_1) \approx f(x_0) + f'(x_0)\delta_{x_1}$ is what is known as the tangent line approximation to the function $$f$$.
• Such an approximation exists when $$f$$ is sufficiently smooth and is accurate when $$\delta_{x_1}$$ is small, so that the difference of $$x_1$$ from the fixed value $$x_0$$ is small.
• We can see graphically how the approximation becomes worse as we take $$\delta_{x_1}$$ too large.

### Concepts in analytical differentiation

• More generally, the tangent line approximation is one kind of general Taylor approximation.

• Suppose we have a point $$x_0$$ fixed, and define $$x_1$$ as a small perturbation $x_1 = x_0+\delta_{x_1},$

• If a function $$f$$ has $$k$$ continuous derivatives we can write $f(x_1) = f(x_0) + f'(x_0)\delta_{x_1} + \frac{f''(x_0)}{2!}\delta_{x_1}^2 + \cdots + \frac{f^{(k)}(x_0)}{k!} \delta_{x_1}^k + \mathcal{o}\left(\delta_{x_1}^{k+1}\right)$

• The $$\mathcal{o}\left(\delta_{x_1}^{k+1}\right)$$ refers to terms in the remainder, that grows or shrinks like the size of the perturbation to the power $$k+1$$.

• This is why this approximation works well when $$\delta_{x_1}$$ is a small perturbation.
• Another important practical example of using this Taylor approximation, when the function $$f$$ has two continuous derivatives, is $f(x_0 + \delta_x) \approx f(x_0) + f'(x_0)\delta_x + f''(x_0) \frac{\delta_x^2}{2}$ which will be used shortly for obtaining solutions to several kinds of equations.

• Particularly, this is strongly related to our second derivative test from univariate calculus.

## An approach to numerical derivation

• At the moment, we consider how Taylor's expansion can be used at first order again to approximate the derivative.

• Recall, we write

\begin{align} f(x_1) &= f(x_0) + f'(x_0) \delta_{x_1} + o\left( \delta_{x_1}^2\right) \\ \Leftrightarrow \frac{f(x_1) - f(x_0)}{ \delta_{x_1}} &= f'(x_0) + o\left( \delta_{x_1}\right) \end{align}

• This says that for a small value of $$\delta_{x_1}$$, we can obtain the numerical approximation of $$f'(x_0)$$ up to approximately to the accuracy of the largest decimal place of $$\delta_{x_1}$$ by the difference on the left hand side.

• This simple approach is the basic version of a finite difference equation, and approximation to the derivative.

• In simple cases this can be sufficiently accurate, variations can give better approximations.

• We will return on how to compute such numerical derivatives in R when we introduce this in full generality of multiple variables.

## Newtwon's method in one variable

• We have seen earlier the basic linear inverse problem,

\begin{align} \mathbf{A}\mathbf{x} = \mathbf{b} \end{align} where $$\mathbf{b}$$ is an observed quantity and $$\mathbf{x}$$ are the unknown variables related to $$\mathbf{b}$$ by the relationships in $$\mathbf{A}$$.

• We observed that a unique solution exists when the $$\mathrm{det}\left(\mathbf{A}\right)\neq 0$$, i.e., all the relationships expressed by the columns (or eigenvalues) are unique.
• A similar problem exists when the relationship between $$\mathbf{x}$$ and $$\mathbf{b}$$ is non-linear, but we still wish to find some such $$\mathbf{x}$$.

• Suppose we know the nonlinear function $$f$$ that gives a relationship in one variable as \begin{align} f(x^\ast) = b \end{align} for an observed $$b$$ but an unknown $$x^\ast$$.

• We will start by re-writing the equation into a more general form, define a function \begin{align} \tilde{f}(x) = f(x)-b. \end{align}

• Thus solving the nonlinear inverse problem in one variable is equivalent to finding the appropriate $$x^\ast$$ for which \begin{align} \tilde{f}(x^\ast)= 0 . \end{align}

• The means of finding one such $$x^\ast$$ is known as root finding.

• The Newton-Raphson method (often Newton's for short) is one classical approach which has inspired many modern techniques for complex systems of equations – we will introduce the main concepts here.

### Newtwon's method in one variable

• We are searching for the point $$x^\ast\in \mathbb{R}$$ for which the modified equation $$\tilde{f}\left(x^\ast\right) = 0$$, and we suppose we have a good initial guess $$x_0$$.
• We define the tangent approximation as, $t(\delta_x) = \tilde{f}(x_0) + \tilde{f}'(x_0) \delta_x$ for some small perturbation value of $$\delta_x$$.
• Recall, $$\tilde{f}'(x_0)$$ refers to the value of the derivative of $$\tilde{f}$$ at the point $$x_0$$ – suppose this value is nonzero.
• In this case, we will examine where the tangent line intersects zero to find a better approximation of $$x^\ast$$.
• Suppose that for $$\delta_{x_0}$$ we have $\begin{matrix} t(\delta_{x_0}) = 0 & \Leftrightarrow & 0= \tilde{f}(x_0) + \tilde{f}'(x_0) \delta_{x_0} & \Leftrightarrow &\delta_{x_0} = \frac{-\tilde{f}(x_0)}{\tilde{f}'(x_0)} \end{matrix}$
• The above solution makes sense as long as $$f'(x_0)$$ is not equal to zero;
• if not, this says that the tangent line intersects zero at $$x_1 = x_0 - \delta_{x_0}$$, giving a new approximation of $$x^\ast$$. Courtesy of Ralf Pfeifer, CC BY-SA 3.0, via Wikimedia Commons

• The process of recursively solving for a better approximation of $$x^\ast$$ terminates when we reach a certain tolerated level of error in the solution or the process times out, failing to converge.
• This method has a direct analog in multiple variables, for which we will need to return to the concept of the matrix inverse.
• We will return to this at the end of the lecture and for now consider a simple example.

### Newtwon's method in one variable – example

• Newton's method in a single variable is implemented by the function uniroot with syntax as
uniroot(function_to_root_find, interval_to_search_for_roots)

• We will consider the polynomial $$x^2-4 = (x+2)(x-2)$$ which clearly has roots at $$\pm 2$$,
f <- function(x){
return (x^2 - 4)
}
uniroot(f, c(-3, 0))

$root  -2.000001$f.root
 3.223832e-06

$iter  6$init.it
 NA

$estim.prec  6.103516e-05  • Notice this solves for the root $$-2$$ in the interval. ### Newtwon's method in one variable – example • Now consider, uniroot(f, c(0, 3))  $root
 2.000001

$f.root  3.223832e-06$iter
 6

$init.it  NA$estim.prec
 6.103516e-05


### Newtwon's method in one variable – example

• But if we try the following interval,
uniroot(f,c(-3,3))

Error in uniroot(f, c(-3, 3)): f() values at end points not of opposite sign

• we get an error message.

• This is because the Newton algorithm needs a good first guess, and here the values of $$f(-3)$$ and $$f(3)$$ are of the same sign

• in this case, the solver doesn't have enough information to begin a search with an initial $$x_0$$ and the interval should be shortened around the first proposal.

## Multiple variables

• Recall our earlier expression f
f <- expression(x^3 * cos(x))

• Q: suppose we differentiate the expression with respect to y, what will be the answer?

• A: all values in the above expression are constant with respect to the value y so that,
D(f,"y")

 0

• This is the basic principle with respect to partial derivatives: expressions that do not include a different variable, e.g., y can be held as constants with the derivative in the other variable.

### Multiple variables

• Suppose we redefine our expression f as,
f <- expression(x^3 * cos(x) * y + y)

• Q: what will the derivative of the above expression evaluate to when take with respect to y? What about with respect to x?

• A: we find that,
D(f, "y")

x^3 * cos(x) + 1

D(f, "x")

(3 * x^2 * cos(x) - x^3 * sin(x)) * y

• We can extend into arbitrary functions,

\begin{align} f:\mathbb{R}^n& \rightarrow \mathbb{R} \\ (x_1, x_2, \cdots, x_n) & \rightarrow f(x_1, x_2, \cdots, x_n) \\ \mathbf{x} & \rightarrow f(\mathbf{x}) \end{align}

• The notation $$\partial_{x_i}$$ refers to the derivative with respect to the variable $$x_i$$ in the same sense as discussed in the above D(f, x) and D(f, y) example.