Commands available in CQT
This page describes all commands that are available in the CQT Toolbox; documentation for each of these commands is available by clicking on their name.
Contents
Arithmetic operations
- C = A + B and C = A - B compute the sum and differences of QT matrices.
- C = A*B computes the matrix product between A and B.
- C = A \ B computes the action of the inverse of A on B. Note that B may be a vector obtained with B = cqt(v) (represented as a QT matrix with zero symbol), which allows to solve linear systems with QT matrices. See LinearSystems for further details.
General commands
- cond: The cond command computes (or estimates) the condition number of a QT matrix.
- conj: Returns the entry-wise complex conjugate of a QT matrix.
- correction: Extracts the correction part of a QT matrix.
- cqtinfo: Prints brief information on a QT matrix.
- cqt: Constructs a new QT matrix.
- cqtoption sets global options for the QT toolbox.
- cqtrank: Returns the rank of the compact correction.
- ctranspose: Returns the complex-conjugate-transpose of the QT matrix $A$.
- diag returns the a QT matrix that only keeps diagonal entries from the original one.
- eye constructs an identity matrix in QT format.
- full returns the full dense version of a QT matrix of finite size.
- inv returns the inverse of a QT matrix, again in the QT format.
- limit computes a vector with the limits of every column for large row indices.
- normest Norm estimator, based on a power method.
- norm Computation of the norm of QT matrices.
- size returns the size of a QT matrix.
- symbol accesses the symbol of the QT matrix under consideration.
- transpose computes the transpose of a QT matrix
- ul computes the UL factorization, that is used (among other things) to solve linear systems and compute the inverse.
Eigenvalue computations
- basins computes the basin of attraction for the matrix iterations used when computing eigenvalues of QT matrices.
- distances computes the distance between the eigenvalues of finite truncations of the infinite-dimensional operator, and its true eigenvalues.
- eig_all computes the spectrum of a QT matrix (decomposed as the continuous part and the isolated eigenvalues).
- eig_single computes a single isolated eigenvalue of a QT matrix starting from an initial approximation.
- range plots the boundary of the continuous spectrum of $A$, which coincides with the symbol evaluated at all the points in the unit circle.
Matrix equations and matrix functions
- cqtlyap solves Lyapunov equations $AX + XA + C = 0$ and Sylvester equations $AX + XB + C = 0$.
- cqtstein solves the Stein equation $AXA + X = C$.
- expm computes the matrix exponential of $A$.
- funm evaluates a generic matrix function by means of a contour integral.
- logm evaluates the matrix logarithm of $A$.
- polyvalm evaluates a polynomial at a matrix $A$.
- sqrtm evaluate the matrix square root of $A$.