## Introduction

This post provides a bit of guidance for students in CSC 115 who are working on the Madhava Graph exercise in Chapter 4 of the course text.

As you will recall, the statement of the problem is as follows:

Recall the function

`madhavaPI()`

from Section 3.4.1. Use this function to write a function called`madhavaGraph()`

that will do the following: given a number \(n\), the function usesggplot2to produce a line graph of the first \(n\) approximations to \(\pi\) using the initial terms of the Madhava series.

The plot should be a line graph similar to the one produced by the

`collatz()`

functions from this Chapter. The function should take a single argument`n`

, whose default value is 10.

It should validate the input: if the number entered is not at least 1, then the function should should explain to the user that the he/she must enter a positive number, and then stop.

## Outline

Based on the specification in the problem, you can set up an initial outline of the desired function, as follows:

```
madhavaGraph <- function(n = 10) {
validate the input: if n isn't at least 1, stop the function
find the sums:
* the sum of the first term (4)
* the sum of the first and second terms (4 - 4/3)
* the sum of the first three terms ( 4 - 4/3 + 4/5)
* and so on until ...
* the sum of the first n terms
make sure these sums are stored in a vector (let's call it "results")
make the plot:
- the vector 1:n (call it "terms") gives the x-coordinates of the points
- the results vector gives y-coordinates
}
```

As you can see, the body of the function has three primary parts:

- validating the input
- finding the sums and storing them in the
`results`

vector - making the plot.

I’ll leave Part 1 entirely to you, but say a bit more about parts 2 and 3.

## Making the Graph

First we’ll think about the graph. Since it is made from functions in the **ggplot2** package, you will need to make sure that you load the package when you are designing the function.^{1}

`library(ggplot2)`

Now suppose that the value of the parameter `n`

is 10. Then by the time you get around to making the graph you’ll have the terms:

```
terms <- 1:n
terms
```

`## [1] 1 2 3 4 5 6 7 8 9 10`

You’ll also have the sums, another vector of length 10:

`results`

```
## [1] 4.000000 2.666667 3.466667 2.895238 3.339683 2.976046 3.283738
## [8] 3.017072 3.252366 3.041840
```

From our study of the Collatz function, we know that the way to make a line graph of these points is:

```
plotTitle <- paste0("The first ", n, " Madhava sums")
p <- ggplot(mapping = aes(x = terms, y = results)) +
geom_point() + geom_line() +
labs( x = "number of terms", y = "Madhava sum",
title = plotTitle)
print(p)
```

Hence we can fill in the outline of our function a bit:

```
madhavaGraph <- function(n = 10) {
validate the input: if n isn't at least 1, stop the function
find the sums:
* the sum of the first term (4)
* the sum of the first and second terms (4 - 4/3)
* the sum of the first three terms ( 4 - 4/3 + 4/5)
* and so on until ...
* the sum of the first n terms
make sure these sums are stored in a vector (let's call it "results")
plotTitle <- paste0("The first ", n, " Madhava sums")
p <- ggplot(mapping = aes(x = terms, y = results)) +
geom_point() + geom_line() +
labs( x = "number of terms", y = "Madhava sum",
title = plotTitle)
print(p)
}
```

## Finding and Storing the Sums

The outline for this part of the task looked like:

- get the sum of the first term (just 1)
- get the sum of the first and second terms (\(4 - 4/3\))
- get the sum of the first three terms (\(4 - 4/3 + 4/5\))
- and so on until …
- get the sum of the first n terms

At each step in above process, we will need to store our sum in the `results`

vector, so that it isn’t lost.

The natural way to accomplish the above task is to use a `for`

-loop in which we iterate by index and store results. The outline of such a loop is:

```
results <- numeric(n)
for ( i in 1:n ) {
results[i] <- the sum of the first i terms
}
```

We already know how to find the sum of the first \(i\) terms: we can just use the function `madhavaPI()`

from Chapter 3:

```
madhavaPI <- function(n = 1000000) {
k <- 1:n
terms <- (-1)^(k+1)*4/(2*k-1)
sum(terms)
}
```

Hence when you are working inside the loop, you can get and store the results all in one line:

`results[i] <- madhavaPI(i)`

Well, to be totally truthful, you can do this *provided that* R knows what `madhavaPI()`

is. Hence you must either:

- hope that whoever uses
`madhavaGraph()`

has`madhavaPI()`

in their Global Environment so that`madhavaGraph()`

can look up the name`madhavaPI`

when it needs it, or - define
`madhavaPI()`

inside the body of`madhavaGraph()`

, before you start computing any sums.

The latter seems to be the safer choice!

## Put It All Together

You still have the task of filling in the outline completely and then testing `madhavaGraph()`

until it works.

The function itself does not need to load

**ggplot2**, but you should make sure to tell anyone who plans to use the function that they need to load the package!↩