# Documentation

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## Single Precision Math

This example shows how to perform arithmetic and linear algebra with single precision data. It also shows how the results are computed appropriately in single-precision or double-precision, depending on the input.

### Create Double Precision Data

Let's first create some data, which is double precision by default.

Ad = [1 2 0; 2 5 -1; 4 10 -1]

1     2     0
2     5    -1
4    10    -1

### Convert to Single Precision

We can convert data to single precision with the single function.

### Create Single Precision Zeros and Ones

We can also create single precision zeros and ones with their respective functions.

n = 1000;
Z = zeros(n,1,'single');
O = ones(n,1,'single');

Let's look at the variables in the workspace.

whos A Ad O Z n
Name         Size            Bytes  Class     Attributes

A            3x3                36  single
O         1000x1              4000  single
Z         1000x1              4000  single
n            1x1                 8  double

We can see that some of the variables are of type single and that the variable A (the single precision version of Ad) takes half the number of bytes of memory to store because singles require just four bytes (32-bits), whereas doubles require 8 bytes (64-bits).

### Arithmetic and Linear Algebra

We can perform standard arithmetic and linear algebra on singles.

B = A'    % Matrix Transpose
B = 3x3 single matrix

1     2     4
2     5    10
0    -1    -1

whos B
Name      Size            Bytes  Class     Attributes

B         3x3                36  single

We see the result of this operation, B, is a single.

C = A * B % Matrix multiplication
C = 3x3 single matrix

5    12    24
12    30    59
24    59   117

C = A .* B % Elementwise arithmetic
C = 3x3 single matrix

1     4     0
4    25   -10
0   -10     1

X = inv(A) % Matrix inverse
X = 3x3 single matrix

5     2    -2
-2    -1     1
0    -2     1

I = inv(A) * A % Confirm result is identity matrix
I = 3x3 single matrix

1     0     0
0     1     0
0     0     1

I = A \ A  % Better way to do matrix division than inv
I = 3x3 single matrix

1     0     0
0     1     0
0     0     1

E = eig(A) % Eigenvalues
E = 3x1 single column vector

3.7321
0.2679
1.0000

F = fft(A(:,1)) % FFT
F = 3x1 single column vector
7.0000 + 0.0000i
-2.0000 + 1.7321i
-2.0000 - 1.7321i

S = svd(A) % Singular value decomposition
S = 3x1 single column vector

12.3171
0.5149
0.1577

P = round(poly(A)) % The characteristic polynomial of a matrix
P = 1x4 single row vector

1    -5     5    -1

R = roots(P) % Roots of a polynomial
R = 3x1 single column vector

3.7321
1.0000
0.2679

Q = conv(P,P) % Convolve two vectors
Q = 1x7 single row vector

1   -10    35   -52    35   -10     1

R = conv(P,Q)
R = 1x10 single row vector

1   -15    90  -278   480  -480   278   -90    15    -1

stem(R); % Plot the result

### A Program that Works for Either Single or Double Precision

Now let's look at a function to compute enough terms in the Fibonacci sequence so the ratio is less than the correct machine epsilon (eps) for datatype single or double.

% How many terms needed to get single precision results?
fibodemo('single')
ans = 19
% How many terms needed to get double precision results?
fibodemo('double')
ans = 41
% Now let's look at the working code.
type fibodemo
function nterms = fibodemo(dtype)
%FIBODEMO Used by SINGLEMATH demo.
% Calculate number of terms in Fibonacci sequence.

% Copyright 1984-2014 The MathWorks, Inc.

fcurrent = ones(dtype);
fnext = fcurrent;
goldenMean = (ones(dtype)+sqrt(5))/2;
tol = eps(goldenMean);
nterms = 2;
while abs(fnext/fcurrent - goldenMean) >= tol
nterms = nterms + 1;
temp  = fnext;
fnext = fnext + fcurrent;
fcurrent = temp;
end
% Notice that we initialize several of our variables, |fcurrent|,
% |fnext|, and |goldenMean|, with values that are dependent on the
% input datatype, and the tolerance |tol| depends on that type as
% well.  Single precision requires that we calculate fewer terms than
% the equivalent double precision calculation.