# The Yorick Programming Language

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Yorick is an interpreted language for numerical analysis used by scientists on machines from Linux laptops to Cray supercomputers.

Yorick has a compact and sophisticated mechanism for describing array indexing and operations, which are used to precisely specify the desired operation to the interpreter. Applying an operation to an array causes the operation to be applied to each element of the array. For example:

```> a = [1,2,3,4,5]
> sqrt(a)
[1,1.41421,1.73205,2,2.23607]
```

What about multiplying two vectors? The default is to perform an element by element multiplication.

```> b = [2,4,6,8,10]
> a*b
[2,8,18,32,50]
```
Those of you who remember physics or linear algebra will recall inner and outer products. The inner product is defined as the sum of the pairwise products:
```> a(+)*b(+)
110
```
The outer product creates a matrix out of each possible multiplication:
```> a(-,)*b(,-)
[[2,4,6,8,10],
[4,8,12,16,20],
[6,12,18,24,30],
[8,16,24,32,40],
[10,20,30,40,50]]
```
The + and - symbols, used where an index would be placed, are called special subscripts and provide precise control over how array operations are executed. The + is the matrix multiplication pseudo-index, which indicates to Yorick along which dimension the addition part of a matrix multiply should be performed. The - is a pseudo-index, creating an index where one did not exist before.

The rank-reducing operators sum, min, max and avg can be used in place of indices.

```> a(max)
5
> b(avg)
6
```

One might wonder why this is necessary, when the equivalent function operators (i.e., min() or avg()) exist? The reason is that for matrices of rank 2 or greater, the rank-reducing index operators allow you to specify exactly how to perform the operation. For example, given a 3x3 array, do you want to average across rows, columns or the entire array?

```> c = [[1,2,3],[4,5,6],[7,8,9]]
> dimsof(c)
[2,3,3]
> avg(c)
5
> c(avg,avg)
5
> c(avg,)
[2,5,8]
> c(,avg)
[4,5,6]
```
Here we have also introduced the dimsof() function operator, which reports the dimensions of the argument. In this case, the result tells us that c is an array of rank 2 with three elements in each direction.

Graphics Operations

Under Linux, Yorick is linked with the GIST graphics subsystem, allowing immediate display of plots and diagrams. Plots are interactive, allowing the user to zoom in and out, stretch axes, and crop the displays using the mouse. Yorick is capable of displaying sequences of plots over time as in a movie, and because of this, the command to prepare for a new image is fma or frame advance.

To plot the value of a function at evenly spaced points, we must first create the x values:

```> x = span(0,10,256)
> dimsof(x)
[1,256]
```

x is now a 256-element array with values that range from 0 to 10.

Figure 1. x-y Plot

The plg function, given vectors for the x and y values, plots x-y graphs.

```plg, sin(x^2), x
```

The results of this command are shown in Figure 1. Note that the arguments are supplied y,x (not x,y). This allows Yorick to supply a default x vector (ranging from 1 to the number of y points), if desired.

Parametric plots are also supported. Consider the following commands which produced the spiral in Figure 2:

```&GT; window, style="vgbox.gs"
&GT; a = span(0,20,256)
&GT; x = a * sin(a)
&GT; y = a * cos(a)
&GT; plg, y, x
```

Figure 2. Spiral Plot

Surface plots are also available, either as a wire frame as in Figure 3:

```&GT; #include "plwf.i"
> orient3
> x = span(-pi,pi,32)(,-:1:32)
> y = transpose(x)
> fma
> plwf, sin(x)*cos(y)
```

Figure 3. Wire Frame Surface Plot

Or a shaded surface rendition as in Figure 4:

```> fma