rat

Rational fraction approximation (continued fraction)

Syntax

R = rat(X)

R = rat(X,tol)

[N,D] = rat(___)

___ = rat(___,Name,Value)

Description

R = rat(X) returns the rational fraction approximation of X to within the default tolerance, 1.e-6*norm(X(:),1). The approximation is a character array containing the simple continued fraction with finite terms.

R = rat(X,tol) approximates X to within the tolerance, tol.

[N,D] = rat(___) returns two arrays, N and D, such that N./D approximates X. You can use this output syntax with any of the previous input syntaxes.

___ = rat(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments to approximate X.

Examples

Find Rational Approximation of Irrational Number

Declare the irrational number $$\sqrt{3}$$ as a symbolic number.

X = sqrt(sym(3))

X = $$\sqrt{3}$$

Find the rational fraction approximation (truncated continued fraction) of this number. The resulting expression is a character vector.

R = rat(X)

R = 
'2 + 1/(-4 + 1/(4 + 1/(-4 + 1/(4 + 1/(-4)))))'

Display the symbolic formula from the character vector R.

displayFormula(["'A rational approximation of X is'"; R])

$$\mathrm{A rational approximation of X is}$$

$$2+\frac{1}{-4+\frac{1}{4+\frac{1}{-4+\frac{1}{4+\frac{1}{-4}}}}}$$

Approximate Value of π with Different Precision

Represent the mathematical quantity $$\pi$$ as a symbolic constant. The constant $$\pi$$ is an irrational number.

X = sym(pi)

X = $$\pi$$

Use vpa to show the decimal representation of $$\pi$$ with 12 significant digits.

Xdec = vpa(X,12)

Xdec = $$3.14159265359$$

Find the rational fraction approximation of $$\pi$$ using the rat function with default tolerance. The resulting expression is a character vector.

R = rat(sym(pi))

R = 
'3 + 1/(7 + 1/(16))'

Use str2sym to turn the character vector into a single fractional number.

Q = str2sym(R)

Q = 
$$\frac{355}{113}$$

Show the decimal representation of the fractional number $$355/113$$. This approximation agrees with $$\pi$$ to 6 decimal places.

Qdec = vpa(Q,12)

Qdec = $$3.14159292035$$

You can specify a tolerance for additional accuracy in the approximation.

R = rat(sym(pi),1e-8)

R = 
'3 + 1/(7 + 1/(16 + 1/(-294)))'

Q = str2sym(R)

Q = 
$$\frac{104348}{33215}$$

The resulting approximation, $$104348/33215$$, agrees with $$\pi$$ to 9 decimal places.

Qdec = vpa(Q,12)

Qdec = $$3.14159265392$$

Approximate Solution of Equation

Solve the equation $$\cos (x)+x^2+x=42$$ using vpasolve. The solution is returned in decimal representation.

syms x
sol = vpasolve(cos(x) + x^2 + x == 42)

sol = $$5.9274875551262136192212919837749$$

Approximate the solution as a continued fraction.

R = rat(sol)

R = 
'6 + 1/(-14 + 1/(5 + 1/(-5)))'

To extract the coefficients in the denominator of the continued fraction, you can use the regexp function and convert them to a character array.

S = char(regexp(R,'(-*\d+','match'))

S = 3x4 char array
    '(-14'
    '(5  '
    '(-5 '

Return the result as a symbolic array.

coeffs = sym(S(:,2:end))

coeffs = 
$$\left(\begin{array}{c}-14\\ 5\\ -5\end{array}\right)$$

Use str2sym to turns the continued fraction R into a single fractional number.

Q = str2sym(R)

Q = 
$$\frac{1962}{331}$$

You can also return the numerator and denominator of the rational approximation by specifying two output arguments for the rat function.

[N,D] = rat(sol)

N = $$1962$$

D = $$331$$

Find Rational Approximation of Golden Ratio

Define the golden ratio $$X=(1+\sqrt{5})/2$$ as a symbolic number.

X = (sym(1) + sqrt(5))/ 2

X = 
$$\frac{\sqrt{5}}{2}+\frac{1}{2}$$

Find the rational approximation of $$X$$ within a tolerance of 1e-4.

R = rat(X,1e-4)

R = 
'2 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3)))))'

To return the rational approximation with 10 coefficients, set the 'Length' option to 10. This option ignores the specified tolerance in the approximation.

R10 = rat(X,1e-4,'Length',10)

R10 = 
'2 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3)))))))))'

To return the rational approximation with all positive coefficients, set the 'Positive' option to true.

Rpos = rat(X,1e-4,'Positive',true)

Rpos = 
'1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1))))))))))'

Input Arguments

X — Input

number | vector | matrix | array | symbolic number | symbolic array

Input, specified as a number, vector, matrix, array, symbolic number, or symbolic array.

Data Types: single | double | sym
Complex Number Support: Yes

tol — Tolerance

scalar

Tolerance, specified as a scalar. N and D approximate X, such that N./D - X < tol. The default tolerance is 1e-6*norm(X(:),1).

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Length',5,'Positive',true

'Length' — Number of coefficients

positive integer

Number of coefficients or terms of the continued fraction, specified as a positive integer. Specifying this option overrides the tolerance argument tol.

Example: 5

'Positive' — Option to return positive coefficients

logical 0 (false) (default) | logical value

Option to return positive coefficients, specified as a logical value (boolean). If you specify true, then rat returns a regular continued fraction expansion with all positive integers in the denominator.

Example: true

Output Arguments

R — Continued fraction

character array

Continued fraction, returned as a character array.

• If X is an array of m elements and all elements are real numbers, then R is returned as a character array with m rows.

• If X is an array of m elements that contains a complex number, then R is returned as a character array with 2m+1 rows. The first m rows of R represent the continued fraction expansion of the real parts of X, followed by ' +i* ... ' in the (m+1)-th row, and the last m rows represent the continued fraction expansions of the imaginary parts of X.

N — Numerator

number | vector | matrix | array | symbolic number | symbolic array

Numerator, returned as a number, vector, matrix, array, symbolic number, or symbolic array. N./D approximates X.

D — Denominator

number | vector | matrix | array | symbolic number | symbolic array

Denominator, returned as a number, vector, matrix, array, symbolic number, or symbolic array. N./D approximates X.

Limitations

• You can only specify the Name,Value arguments, such as 'Length',5,'Positive',true, if the array X contains a symbolic number or the data type of X is sym.

More About

Simple Continued Fraction

The rat function approximates each element of X by a simple continued fraction of the form

$$R=\frac{N}{D}=a_1+\frac{1}{a_2+\frac{1}{\ddots +\frac{1}{a_k}}}$$

with a finite number of integer terms $$a_1,a_2,\dots ,a_k$$. The accuracy of the rational approximation increases with the number of terms.