## MATLAB |

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.

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

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

Ad = 1 2 0 2 5 -1 4 10 -1

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).

We can perform standard arithmetic and linear algebra on singles.

```
B = A' % Matrix Transpose
```

B = 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 = 5 12 24 12 30 59 24 59 117

```
C = A .* B % Elementwise arithmetic
```

C = 1 4 0 4 25 -10 0 -10 1

```
X = inv(A) % Matrix inverse
```

X = 5 2 -2 -2 -1 1 0 -2 1

```
I = inv(A) * A % Confirm result is identity matrix
```

I = 1 0 0 0 1 0 0 0 1

```
I = A \ A % Better way to do matrix division than inv
```

I = 1 0 0 0 1 0 0 0 1

```
E = eig(A) % Eigenvalues
```

E = 3.7321 0.2679 1.0000

```
F = fft(A(:,1)) % FFT
```

F = 7.0000 + 0.0000i -2.0000 + 1.7321i -2.0000 - 1.7321i

```
S = svd(A) % Singular value decomposition
```

S = 12.3171 0.5149 0.1577

```
P = round(poly(A)) % The characteristic polynomial of a matrix
```

P = 1 -5 5 -1

```
R = roots(P) % Roots of a polynomial
```

R = 3.7321 1.0000 0.2679

```
Q = conv(P,P) % Convolve two vectors
R = conv(P,Q)
```

Q = 1 -10 35 -52 35 -10 1 R = 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') % How many terms needed to get double precision results? fibodemo('double') % Now let's look at the working code. type fibodemo % 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.

ans = 19 ans = 41 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