Writing Fast Matlab Code #7: Referencing Operations
Referencing in Matlab is varied and powerful enough to deserve a section of discussion. Good under- standing of referencing enables vectorizing a broader range of programming situations.
Subscripts vs. Indices
Subscripts are the most common method used to refer to matrix elements, for example, A(3,9) refers to row 3, column 9. Indices are an alternative referencing method. Consider a 10 × 10 matrix A. Internally, Matlab stores the matrix data linearly as a one-dimensional, 100-element array of data in column-major order.
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An index refers to an element’s position in this one-dimensional array. Using an index, A(83) also refers to the element on row 3, column 9.
It is faster to scan down columns than over rows. Column-major order means that elements along a column are sequential in memory while elements along a row are further apart. Scanning down columns promotes cache efficiency.
Conversion between subscripts and indices can be done with the sub2ind and ind2sub functions. However, because these are M-file functions rather than fast built-in operations, it is more efficient to compute conversions directly (also see Section 6). For a two-dimensional matrix A of size M×N, the conversions between subscript (i,j) and index (index) are
A(i,j) ↔ A(i + (j-1)*M) A(index) ↔ A(rem(index-1,M)+1, floor((index-1)/M)+1)
Indexing extends to N-D matrices as well, with indices increasing first through the columns, then through the rows, through the third dimension, and so on. Subscript notation extends intuitively,
A(. . . , dim4, dim3, row, col).
Vectorized Subscripts
It is useful to work with submatrices rather than individual elements. This is done with a vector of indices or subscripts. If A is a two-dimensional matrix, a vector subscript reference has the syntax
A(rowv, colv)
where rowv and colv are vectors. Both may be of any length and their elements may be in any order. If either is a matrix, it is reshaped to a vector. There is no difference between using row vectors or column vectors in vector subscripts.
Let M = length(rowv) and N = length(colv). Then a vector subscripted matrix on the right-hand side∗ returns a submatrix of size M×N:
A vector subscripted matrix can also appear on the left-hand side as
A(rowv, colv) = scalar or M×N matrix
If any elements in the destination reference are repeated, the source element with the greater index dominates. For example,
>> A = zeros(2); A([1,2],[2,2]) = [1,2;3,4]
A =
0 2
0 4
Vector subscript references extend intuitively in higher dimensions.
∗ The left- and right-hand sides (LHS and RHS) refer to the two sides of an assignment, “LHS = RHS.”
Vector Indices
Multiple elements can also be referenced with vector indices.
A(indexv)
where indexv is an array of indices. On the right-hand side, a vector index reference returns a matrix the same size and shape as indexv. If indexv is a 3 × 4 matrix, then A(indexv) is the 3 × 4 matrix
While vector subscripts are limited to referring to block-shaped submatrices, vector indices can refer to any subset of elements.
If a vector index reference is on the left-hand side, the right-hand side must return a matrix of the same size as indexv or a scalar. As with vector subscripts, ambiguous duplicate assignments use the value with greater source index.
Reference Wildcards
Using the wildcard, :, in a subscript refers to an entire row or column. For example, A(:,1) refers to every row in column one–the entire first column. This can be combined with vector subscripts, A([2,4],:) refers to the second and fourth rows.
When the wildcard is used in a vector index, the entire matrix is referenced. On the right-hand side, this always returns a column vector.
A(:) = column vector
This is frequently useful: for example, if a function input must be a row vector, the user’s input can be quickly reshaped into row vector with A(:).’ (make a column vector and transpose to a row vector).
A(:) on the left-hand side assigns all the elements of A, but does not change its size. For example,
A(:) = 8 changes all elements of matrix A to 8.
Logical Indexing
Given a logical array mask with the same size as A,
A(mask)
refers to all elements of A where the corresponding element of mask is 1 (true). It is equivalent to A(find(mask)). A common usage of logical indexing is to refer to elements based on a per-element condition, for example,
A(abs(A) < 1e−3) = 0
sets all elements with magnitude less than 10−3 to zero. Logical indexing is also useful to select non- rectangular sets of elements, for example,
A(logical(eye(size(A))))
references the diagonal elements of A. Note that for a right-hand side reference, it is faster to use
diag(A) to get the diagonal of A.
Deleting Submatrices with [ ]
Elements in a matrix can be deleted by assigning the empty matrix. For example, A([3,5]) = [ ] deletes the third and fifth element from A. If this is done with index references, the matrix is reshaped into a row vector.
It is also possible to delete with subscripts if all but one subscript are the wildcard.
A(2,:) = [ ]
deletes the second row.
Deletions like A(2,1) = [ ] or even A(2,1:end) = [ ] are illegal.
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