# 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]

`Ad = `*3×3*
1 2 0
2 5 -1
4 10 -1

### Convert to Single Precision

We can convert data to single precision with the `single`

function.

`A = single(Ad); % or A = cast(Ad,'single');`

### 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 Ad 3x3 72 double 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.