MATLAB Function Reference
rat, rats

Rational fraction approximation

Syntax

• ```[N,D] = rat(X)
[N,D] = rat(X,tol)
rat(...)
S = rats(X,strlen)
S = rats(X)
```

Description

Even though all floating-point numbers are rational numbers, it is sometimes desirable to approximate them by simple rational numbers, which are fractions whose numerator and denominator are small integers. The `rat` function attempts to do this. Rational approximations are generated by truncating continued fraction expansions. The `rats` function calls `rat`, and returns strings.

```[N,D] = rat(X) ``` returns arrays `N` and `D` so that `N./D` approximates `X` to within the default tolerance, `1.e-6*norm(X(:),1)`.

```[N,D] = rat(X,tol) ``` returns `N./D` approximating `X` to within `tol`.

```rat(X), ``` with no output arguments, simply displays the continued fraction.

```S = rats(X,strlen) ``` returns a string containing simple rational approximations to the elements of `X`. Asterisks are used for elements that cannot be printed in the allotted space, but are not negligible compared to the other elements in `X`. `strlen` is the length of each string element returned by the `rats` function. The default is `strlen` `=` `13`, which allows 6 elements in 78 spaces.

```S = rats(X) ``` returns the same results as those printed by MATLAB with `format rat`.

Examples

Ordinarily, the statement

• ``` s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7
```

produces

• ``` s =
0.7595
```

However, with

• ``` format rat
```

or with

• ``` rats(s)
```

the printed result is

• ``` s =
319/420
```

This is a simple rational number. Its denominator is 420, the least common multiple of the denominators of the terms involved in the original expression. Even though the quantity `s` is stored internally as a binary floating-point number, the desired rational form can be reconstructed.

To see how the rational approximation is generated, the statement `rat(s)`

produces

• ``` 1 + 1/(-4 + 1/(-6 + 1/(-3 + 1/(-5))))
```

And the statement

• ``` [n,d] = rat(s)
```

produces

• ```n = 319`,` d = 420
```

The mathematical quantity is certainly not a rational number, but the MATLAB quantity `pi` that approximates it is a rational number. `pi` is the ratio of a large integer and `2`52`:`

• ``` 14148475504056880/4503599627370496
```

However, this is not a simple rational number. The value printed for `pi` with `format` `rat`, or with `rats(pi)`, is

• ``` 355/113
```

This approximation was known in Euclid's time. Its decimal representation is

• ``` 3.14159292035398
```

and so it agrees with `pi` to seven significant figures. The statement

• ``` rat(pi)
```

produces

• ``` 3 + 1/(7 + 1/(16))
```

This shows how the `355/113` was obtained. The less accurate, but more familiar approximation `22/7` is obtained from the first two terms of this continued fraction.

Algorithm

The `rat(X)` function approximates each element of `X` by a continued fraction of the form

The s are obtained by repeatedly picking off the integer part and then taking the reciprocal of the fractional part. The accuracy of the approximation increases exponentially with the number of terms and is worst when `X` `=` `sqrt(2)`. For `x` `=` `sqrt(2)`, the error with `k` terms is about `2.68*(.173)^k`, so each additional term increases the accuracy by less than one decimal digit. It takes 21 terms to get full floating-point accuracy.

`format`