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Matrix functions
Currently, the only functions in this category are max and min (most element-wise functions are listed under scientific functions later, and element-wise operators are list under Matrix algebra). max(A,B) for equal-sized A and B computes the element-wise max of the matrices. Just as for matrix operators, matrix functions also work in these cases:
- A or B is a scalar, in which case that value is applied against all values in the other matrix.
- A or B is a 1 x 1 matrix, in which case the value is treated as a scalar as above.
- A or B is a vector whose length matches the corresponding dimension of the other matrix. In that case, each value in the vector is applied against all the elements in the other matrix in the corresponding row or column.
Reducing functions apply a binary operator cumulatively to rows or columns of the input matrix, and produce either a vector output (standard reducing functions) or a same-size matrix (cumulative reducing functions). The standard reducing functions are
- maxi(A, n), mini(A, n), sum(A, n), mean(A, n), variance(A, n), sdev(A, n)
All of these functions can be called without a dimension argument, which is a convenience form intended for vector arguments. So e.g. sum(A) for a vector A reduces along the long dimension of A. Calling sum(A) without the dimension argument for a non-vector A will reduce along dimension 1.
The cumulative reducing functions right now are:
- cumsum(A, n)
The following function constructs dense matrices from tuples of indices.
- accum(inds:IMat, vals:DMat, nrows, ncols)
The sort functions sorts columns of a matrix in ascending order. sort2(A) returns both the a sorted version of A and a permutation matrix P such that for each i, B(?,i) = A(P(?,i),i). Or if we add column offsets to P, POFF, then we have that B=A(P+POFF).
scala> a res16: BIDMat.DMat = 0.14771 0.53622 0.85483 0.042617 0.96534 0.41199 0.63428 0.014676 0.66535 0.029090 0.85247 0.61777 0.38885 0.50339 0.81829 0.77564 scala> val (b, p) = sort2(a) b: BIDMat.DMat = 0.14771 0.029090 0.63428 0.014676 0.38885 0.41199 0.81829 0.042617 0.66535 0.50339 0.85247 0.61777 0.96534 0.53622 0.85483 0.77564 p: BIDMat.IMat = 0 2 1 1 3 1 3 0 2 3 2 2 1 0 0 3 scala> val poff = iones(4,1)*irow(0 to 12 by 4) poff: BIDMat.IMat = 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12 scala> a(p + poff) res21: BIDMat.DMat = 0.14771 0.029090 0.63428 0.014676 0.38885 0.41199 0.81829 0.042617 0.66535 0.50339 0.85247 0.61777 0.96534 0.53622 0.85483 0.77564 scala> b res22: BIDMat.DMat = 0.14771 0.029090 0.63428 0.014676 0.38885 0.41199 0.81829 0.042617 0.66535 0.50339 0.85247 0.61777 0.96534 0.53622 0.85483 0.77564
sort and sort2 always sort in ascending order. For descending sorts, there are functions sortdown and sortdown2.
To sort each row, use sort(A,2) or sortdown(A,2).
To sort rows of a matrix (in lexicographic order), there are two functions sortlex(A) and sortrows(A). sortlex returns only the permutation matrix for the sort. sortrows returns both the permutation matrix and the matrix itself.
scala> a
a: BIDMat.DMat =
0.083308 0.0053793 0.60398
0.75904 0.91294 0.43385
0.53446 0.91638 0.59250
0.39092 0.65965 0.020080
scala> val (aa, ii) = sortrows(a)
aa: BIDMat.DMat =
0.083308 0.0053793 0.60398
0.39092 0.65965 0.020080
0.53446 0.91638 0.59250
0.75904 0.91294 0.43385
ii: BIDMat.IMat =
0
3
2
1
scala> a(ii,?)
res26: BIDMat.DMat =
0.083308 0.0053793 0.60398
0.39092 0.65965 0.020080
0.53446 0.91638 0.59250
0.75904 0.91294 0.43385
Finally, for descending sorts there are functions sortlexdown and sortrowsdown.
unique and uniquerows
The function inv(A) returns the inverse of the square matrix A. Matrix inversion is an O(n^3) operation.
There are two eigenvalue functions for symmetric matrices. Both can return two values: a column vector of eigenvalues, and a square matrix whose columns are the corresponding eigenvectors. The eig function takes two arguments, the matrix A, and a boolean flag which determines whether the eigenvectors are returned. eig uses reduction to tridiagonal form followed by calculation of eigenvalues. The second function is feig which uses a faster, divide-and-conquer algorithm for eigenvalues and eigenvectors.
scala> aa aa: BIDMat.DMat = 2.9260 1.6648 2.5727 2.2300 1.6648 1.3799 1.6853 1.4514 2.5727 1.6853 3.1019 2.3740 2.2300 1.4514 2.3740 2.2825 scala> val (v, m)= eig(aa, true) v: BIDMat.DMat = 0.27301 0.31132 0.45670 8.6492 m: BIDMat.DMat = -0.13731 0.40833 -0.71388 0.55207 -0.12716 -0.90024 -0.21440 0.35697 -0.50598 0.14970 0.62654 0.57360 0.84200 0.020595 0.22771 0.48863 scala> aa * m /@ m res31: BIDMat.DMat = 0.27301 0.31132 0.45670 8.6492 0.27301 0.31132 0.45670 8.6492 0.27301 0.31132 0.45670 8.6492 0.27301 0.31132 0.45670 8.6492
Both algorithms take O(n^3) steps, and feig is about twice as fast as eig when retrieving the eignvectors. feig should be used only with positive definite matrices. eig requires about twice as many flops (3 n^3 vs 4/3 n^3) as matrix inversion, but the calculation is less uniform and the MFlop/s achieved is lower.
The function chol computes a Cholesky factorization A=L*L^T of a symmetric matrix. The factor is returned a a lower-triangular matrix.
scala> aa res33: BIDMat.DMat = 2.9260 1.6648 2.5727 2.2300 1.6648 1.3799 1.6853 1.4514 2.5727 1.6853 3.1019 2.3740 2.2300 1.4514 2.3740 2.2825 scala> val b = chol(aa) b: BIDMat.DMat = 1.7105 0 0 0 0.97325 0.65778 0 0 1.5040 0.33675 0.85233 0 1.3037 0.27752 0.37518 0.60420 scala> b * b.t res34: BIDMat.DMat = 2.9260 1.6648 2.5727 2.2300 1.6648 1.3799 1.6853 1.4514 2.5727 1.6853 3.1019 2.3740 2.2300 1.4514 2.3740 2.2825
Cholesky factorization is an O(n^3) calculation, and is about twice as fast as matrix inversion.
The trisolve function solves triangular systems of the form T x = y for x. trisolve accepts a string argument that specifies the type of calculation. The mode string has three letters:
- Char1 is "U" or "L" for upper or lower triangular matrices.
- Char2 = "N", "T" or "C" for A not-transposed, transposed or conjugate respectively.
- Char3 = "N" or "U" whether the leading diagonal is non-unit "N" or unit "U" respectively.
scala> b res35: BIDMat.DMat = 1.7105 0 0 0 0.97325 0.65778 0 0 1.5040 0.33675 0.85233 0 1.3037 0.27752 0.37518 0.60420 scala> val y = rand(4,1) y: BIDMat.DMat = 0.047913 0.65338 0.58416 0.26086 scala> val x = trisolve(b, y, "LNN") x: BIDMat.DMat = 0.028011 0.95187 0.25987 -0.22728 scala> b*x res36: BIDMat.DMat = 0.047913 0.65338 0.58416 0.26086
Triangular matrix solving is an O(n^2) operation for vectors x and y, or O(n^2 k) for k x n matrices x, y.