Linear Algebra

Linear Systems

Triangular Systems

solve_Lx_b(L, b)	Solves the system L x = b using back substitution
solve_LTx_b(L, b)	Solves the system L^T x = b using back substitution
solve_Ux_b(U, b)	Solves the system U x = b using back substitution
solve_UTx_b(U, b)	Solves the system U^T x = b using back substitution
solve_spd_chol(L, b)	Solves a symmetric positive definite system A x = b where A = L L’

Special Dense Linear Systems

mult_with_submatrix(A, columns, x)	Computes \(b = A[:, I] x\)
solve_on_submatrix(A, columns, b)	Solves the problem \(A[:, I] x = b\) where I is an index set of selected columns

Singular Value Decomposition

Fundamental Subspaces

orth(A[, rcond])	Constructs an orthonormal basis for the range of A using SVD
row_space(A[, rcond])	Constructs an orthonormal basis for the row space of A using SVD
null_space(A[, rcond])	Constructs an orthonormal basis for the null space of A using SVD
left_null_space(A[, rcond])	Constructs an orthonormal basis for the left null space of A using SVD
effective_rank(A[, rcond])	Returns the effective rank of A based on its singular value decomposition
effective_rank_from_svd(u, s, vh[, rcond])	Returns the effective rank of a matrix from its SVD
singular_values(A)	Returns the singular values of a matrix

SVD for Bidiagonal Matrices

bdsqr(alpha, beta, k)	Computes the SVD of the bidiagonal matrix
bdsqr_jit(alpha, beta, k)	Computes the SVD of the bidiagonal matrix

Truncated SVD

lansvd_simple(A, k, p0)	Returns the k largest singular values and corresponding vectors
lansvd_simple_jit(A, k, p0)	Returns the k largest singular values and corresponding vectors
lanbpro_init(A, k, p0, options)	Initialize the state with a starting vector
lanbpro_iteration(A, state, options)	One single (j-th) iteration of Lanczos bidiagonalization with partial reorthogonalization algorithm
lanbpro_iteration_jit(A, state, options)	One single (j-th) iteration of Lanczos bidiagonalization with partial reorthogonalization algorithm
lanbpro(A, k, p0)	K steps of the Lanczos bidiagonalization with partial reorthogonalization
lanbpro_jit(A, k, p0)	K steps of the Lanczos bidiagonalization with partial reorthogonalization

Orthogonalization

Householder Reflections

householder_vec(x)	Computes a Householder vector for \(x\)
householder_matrix(x)	Computes a Householder refection matrix for \(x\)
householder_premultiply(v, beta, A)	Pre-multiplies a Householder reflection defined by \(v, beta\) to a matrix A, PA
householder_postmultiply(v, beta, A)	Post-multiplies a Householder reflection defined by \(v, beta\) to a matrix A, AP
householder_ffm_jth_v_beta(A, j)	GVL4 EQ 5.1.4 v, beta calculation
householder_ffm_premultiply(A, C)	Computes Q^T C where Q is stored in its factored form in A.
householder_ffm_backward_accum(A, k)	Computes k columns of Q from the factored form representation of Q stored in A.
householder_ffm_to_wy(A)	Computes the WY representation of Q such that Q = I_m - W Y^T from the factored form representation
householder_qr_packed(A)	Computes the QR = A factorization of A using Householder reflections.
householder_split_qf_r(A)	Splits a packed QR factorization into QF and R
householder_qr(A)	Computes the QR = A factorization of A using Householder reflections

Subspaces

Projection

project_to_subspace(U, v)	Projects a vector to a subspace
is_in_subspace(U, v)	Checks whether a vector v is in the subspace spanned by an ONB U or not

Principal Angles

principal_angles_cos(A, B)	Returns the cosines of principal angles between two subspaces
principal_angles_rad(A, B)	Returns the principal angles between two subspaces in radians
principal_angles_deg(A, B)	Returns the principal angles between two subspaces in degrees
smallest_principal_angle_cos(A, B)	Returns the cosine of smallest principal angle between two subspaces
smallest_principal_angle_rad(A, B)	Returns the smallest principal angle between two subspaces in radians
smallest_principal_angle_deg(A, B)	Returns the smallest principal angle between two subspaces in degrees
smallest_principal_angles_cos(subspaces)	Returns the smallest principal angles between each pair of subspaces
smallest_principal_angles_rad(subspaces)	Returns the smallest principal angles between each pair of subspaces in radians
smallest_principal_angles_deg(subspaces)	Returns the smallest principal angles between each pair of subspaces in degrees
subspace_distance(A, B)	Returns the Grassmannian distance between two subspaces

Affine Spaces

Homogeneous Coordinate System

homogenize(X[, value])
homogenize_vec(x[, value])
homogenize_cols(X[, value])

Standard Matrices

Random matrices

gaussian_mtx(key, N, D[, normalize_atoms])

A dictionary/sensing matrix where entries are drawn independently from normal distribution.

Special Matrices

pascal(n[, symmetric])	Returns a pascal matrix of size n imes n
pascal_jit(n[, symmetric])	Returns a pascal matrix of size n imes n

Toeplitz Matrices

toeplitz_mat(c, r)	Constructs a Toeplitz matrix
toeplitz_mult(w, x)	Multiplies a Toeplitz matrix with a vector

Circulant Matrices

circulant_mat(c)	Constructs a circulant matrix
circulant_mult(c, x)	Multiplies a circulant matrix with a vector