A grid is a set of evenly spaced points. You can use SAS to create a grid of points on an interval, in a rectangular region in the plane, or even in higher-dimensional regions like the parallelepiped shown at the left, which is generated by three vectors. You can use vectors, matrices, and functions in the SAS/IML language to obtain a grid in a parallelepiped.

### Evenly spaced points in an interval

In one dimension, you can use a DO loop in the SAS DATA step to construct evenly spaced points in an interval. In the SAS/IML language, you can construct a vector of evenly spaced values operation without writing a DO loop. You can use the DO function, or you can use the colon operator (which SAS/IML documentation calls the "index creation operator"). The results are equivalent, as shown in the following program:

proc iml; a = 2; b = 5; N = 5; delta = (b-a) / N; y = do(a, b, delta); /* Method 1: DO function: start=1, stop=b, step=delta */ x = a + delta*(0:N); /* Method 2: Use colon operator and vector sum */ print y, x; |

### Subdivide a vector subspace

The second method generalizes to any linear subspace. For example, suppose you are given a vector **u** with magnitude α = ||**u**||. Then the following SAS/IML statements create points that are evenly spaced points along the vector **u**.
The magnitudes of the *N* vectors are 0, α/*N*, 2α/*N*, and so forth up to α.

/* linear spacing along a vector */ u = {3 2 1}; delta = u / N; /* Note: delta is vector */ w = delta @ T(0:N); /* i_th row is (i-1)*u */ print w; |

It's always exciting to use the Kronecker product operator (@). The Kronecker product operator multiplies `delta` by 0, 1, 2, ..., *N* and stacks the results. Notice that the *i*th column of the `w` matrix is a linear progression from 0 to the *i*th component of **u**. Each row is a vector in the same direction as **u**.

### Grids in a linear subspace

It is only slightly harder to generate a grid of points in two or higher dimensions. In the DATA step, you can use nested DO loops to generate a grid of points in SAS. In the SAS/IML language, you can use the EXPANDGRID function.

*Generate evenly spaced grid points in SAS. #SAStip*

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In fact, with the help of linear algebra, the EXPANDGRID function enables you to construct a grid in *any* linear subspace. Recall that a linear subspace is the "span," or set of all linear combinations, of *k* basis vectors.

From a linear algebra perspective, a rectangular grid is a set of discrete, evenly spaced, linear combinations of the standard Cartesian basis vectors
**e**_{1},
**e**_{2}, ...
**e**_{k}.
In a linear algebra course you learn that a matrix multiplication enables you to change to another set of basis vectors.

To give a concrete example, consider a rectangular grid of points in the square [0,1]x[0,1].
In the following SAS/IML program, the
rows of the matrix G contain the grid points. You can
think of each row as being a set of coefficients for a linear combination of the
basis vectors
**e**_{1} and
**e**_{2}.
You can use those same coefficients to form linear combinations of any other basis vectors. The program shows how to form a grid in the two-dimensional linear subspace that is spanned by the vectors
**u** = { 3 2 1} and
**v** = {-3 5 2}:

h = 1/N; /* generate linear combinations of basis vectors e1 and e2 */ G = ExpandGrid( do(0,1,h), do(0,1,h) ); u = { 3 2 1}; v = {-3 5 2}; M = u // v; /* create matrix of new basis vectors */ grid = G * M; /* grid = linear combinations of basis vectors u and v */ |

In the graph, the three-dimensional vectors **u** and **v** are indicated. The grid of points is a set of linear combinations of the form (*i*/*N*) **u** + (*j*/*N*) **v**, where 0 ≤ i, j ≤ *N*. The grid lies on a two-dimensional subspace spanned by **u** and **v**. The points in the grid are colored according to the values of the third component: blue points (lower left corner) correspond to low *z*-values whereas red points (upper right corner) correspond to high *z*-values.

Grids are important for visualizing multivariate functions. The function in this example is linear, but grids also enable you to visualize and understand nonlinear transformations.

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