# LinearAlgebra¶

Usage

use LinearAlgebra;


Submodules

Linear Algebra Module

A high-level interface to linear algebra operations and procedures.

Note

This is an early prototype package module. As a result, interfaces may change over the next release.

## Compiling with Linear Algebra¶

The linear algebra module uses the BLAS module. In order to compile a Chapel program with this module, be sure to have a BLAS implementation available on your system. See the BLAS documentation for further details.

## Linear Algebra Interface¶

Matrix and Vector representation

Matrices and vectors are represented as Chapel arrays, which means the convenience constructors provided in this module are not necessary to use this module's functions. This also means that matrices (2D arrays) and vectors (1D arrays) will work with any other Chapel library that works with arrays.

Note

This documentation uses the terms matrix to refer to 2D arrays, and vector to refer to 1D arrays.

Indexing

All functions that return new arrays will return arrays over 0-based indices, unless otherwise specified.

Matrix multiplication

dot is the general matrix multiplication function provided in this module. This function supports any combination of scalars, vectors (1D arrays), and matrices (2D arrays). See the dot documentation for more information.

The dot function, along with others may be given a matrix-specific operator in future releases.

Row vs Column vectors

Row and column vectors are both represented as 1D arrays and are indistinguishable in Chapel. In the dot function, matrix-vector multiplication assumes a column vector, vector-matrix multiplication assumes a row vector, vector-vector multiplication is always treated as an inner-product, as the function name implies. An outer product can be computed with the outer function.

Promotion flattening

Promotion flattening is an unintended consequence of Chapel's promotion feature, when a multi-dimensional array assignment gets type-inferred as a 1-dimensional array of the same size, effectively flattening the array dimensionality. This can result in unexpected behavior in programs such as this:

var A = Matrix(4, 4),
B = Matrix(4, 4);
// A + B is a promoted operation, and C becomes a 1D array
var C = A + B;
// This code would then result in an error due to rank mismatch:
C += A;


To avoid this, you can avoid relying on inferred-types for new arrays:

var A = Matrix(4, 4),
B = Matrix(4, 4);
// C's type is not inferred and promotion flattening is not a problem
var C: [A.domain] A.eltType = A + B;
// Works as expected
C += A;


Alternatively, you can use the LinearAlgebra helper routines, which create and return a new array:

var A = Matrix(4, 4),
B = Matrix(4, 4);
// matPlus will create a new array with an explicit type
var C = matPlus(A, B);
// Works as expected
C += A;


Promotion flattening is not expected to be an issue in future releases.

proc Vector(length, type eltType = real)

Return a vector (1D array) over domain {0..#length}

proc Vector(space: range, type eltType = real)

Return a vector (1D array) over domain {space}

proc Vector(Dom: domain(1), type eltType = real)

Return a vector (1D array) over domain Dom

proc Vector(A: [?Dom] ?Atype, type eltType = Atype)

Return a vector (1D array) with domain and values of A

proc Vector(x: ?t, Scalars ...?n, type eltType)

Return a vector (1D array), given 2 or more numeric values

If type is omitted, it will be inferred from the first argument

proc Matrix(rows, type eltType = real)

Return a square matrix (2D array) over domain {0..#rows, 0..#rows}

proc Matrix(rows, cols, type eltType = real)

Return a matrix (2D array) over domain {0..#rows, 0..#cols}

proc Matrix(space: range, type eltType = real)

Return a square matrix (2D array) over domain {space, space}

proc Matrix(rowSpace: range, colSpace: range, type eltType = real)

Return a matrix (2D array) over domain {rowSpace, colSpace}

proc Matrix(Dom: domain, type eltType = real)

Return a matrix (2D array) over domain Dom

proc Matrix(A: [?Dom] ?Atype, type eltType = Atype)

Return a matrix (2D array) with domain and values of A.

A can be sparse (CS) or dense.

proc Matrix(const Arrays ...?n, type eltType)

Return a matrix (2D array), given 2 or more vectors, such that the vectors form the rows of the matrix. In other words, the vectors are concatenated such that the ith vector corresponds to the matrix slice: A[i, ..]

If type is omitted, it will be inferred from the first array.

For example:

var A = Matrix([1, 2, 3],
[4, 5, 6],
[7, 8, 9]);
/* Produces the 3x3 matrix of integers:
1 2 3
4 5 6
7 8 9
*/

proc eye(m, type eltType = real)

Return a square identity matrix over domain {0..#m, 0..#m}

proc eye(m, n, type eltType = real)

Return an identity matrix over domain {0..#m, 0..#n}

proc eye(Dom: domain(2), type eltType = real)

Return an identity matrix over domain Dom

proc transpose(A: [?Dom] ?eltType)

Transpose vector, matrix, or domain.

Note

Since row vectors and columns vectors are indistinguishable, passing a vector to this function will return that vector unchanged

proc _array.T

Transpose vector or matrix

proc matPlus(A: [?Adom] ?eltType, B: [?Bdom] eltType)

Add matrices, maintaining dimensions, deprecated for _array.plus

proc _array.plus(A: [?Adom])

proc matMinus(A: [?Adom] ?eltType, B: [?Bdom] eltType)

Subtract matrices, maintaining dimensions, deprecated for _array.minus

proc _array.minus(A: [?Adom])

Subtract matrices, maintaining dimensions

proc _array.times(A: [?Adom])

Element-wise multiplication, maintaining dimensions

proc _array.elementDiv(A: [?Adom])

Element-wise division, maintaining dimensions

proc dot(A: [?Adom] ?eltType, B: [?Bdom] eltType)

Generic matrix multiplication, A and B can be a matrix, vector, or scalar.

When A is a vector and B is a matrix, this function implicitly computes dot(transpose(A), B), which may not be as efficient as passing A and B in the reverse order.

proc _array.dot(A: [])

Compute the dot-product

proc inner(A: [?Adom], B: [?Bdom])

Inner product of 2 vectors

proc outer(A: [?Adom] ?eltType, B: [?Bdom] eltType)

Outer product of 2 vectors

proc matPow(A: [], b)

Return the matrix A to the bth power, where b is a positive integral type.

proc cross(A: [?Adom] ?eltType, B: [?Bdom] eltType)

Return cross-product of 3-element vectors A and B with domain of A

proc diag(A: [?Adom] ?eltType, k = 0)

Return a Vector containing the diagonal elements of A if the argument A is of rank 2. Return a diagonal Matrix whose diagonal contains elements of A if argument A is of rank 1.

proc tril(A: [?D] ?eltType, k = 0)

Return lower triangular part of matrix, above the diagonal + k, where k = 0 includes the diagonal, and k = -1 does not include the diagonal. For example:

var A = Matrix(4, 4, eltType=int);
A = 1;

tril(A);
/* Returns:

1    0    0    0
1    1    0    0
1    1    1    0
1    1    1    1
*/

tril(A, 1);
/* Returns:

1    1    0    0
1    1    1    0
1    1    1    1
1    1    1    1
*/

tril(A, -1);
/* Returns:

0    0    0    0
1    0    0    0
1    1    0    0
1    1    1    0
*/

proc triu(A: [?D] ?eltType, k = 0)

Return upper triangular part of matrix, above the diagonal + k, where k = 0 includes the diagonal, and k = 1 does not include the diagonal. For example:

var A = Matrix(4, 4, eltType=int);
A = 1;

triu(A);
/* Returns:

1    1    1    1
0    1    1    1
0    0    1    1
0    0    0    1
*/

triu(A, 1);
/* Returns:

0    1    1    1
0    0    1    1
0    0    0    1
0    0    0    0
*/

triu(A, -1);
/* Returns:

1    1    1    1
1    1    1    1
0    1    1    1
0    0    1    1
*/

proc isDiag(A: [?D] ?eltType)

Return true if matrix is diagonal

proc isHermitian(A: [?D])

Return true if matrix is Hermitian

proc isSymmetric(A: [?D]): bool

Return true if matrix is symmetric

proc isTril(A: [?D] ?eltType, k = 0): bool

Return true if matrix is lower triangular below the diagonal + k, where k = 0 does not include the diagonal, and k = 1 includes the diagonal

proc isTriu(A: [?D] ?eltType, k = 0): bool

Return true if matrix is upper triangular above the diagonal + k, where k = 0 does not include the diagonal, and k = -1 includes the diagonal

proc isSquare(A: [?D])

Return true if matrix is square

proc trace(A: [?D] ?eltType)

Return the trace of square matrix A