AB08NX Construction of a reduced system with input/output matrix Dr of full row rank, preserving transmission zeros
AB09AX Balance & Truncate model reduction with state matrix in real Schur form AB09BX Singular perturbation approximation based model reduction with state matrix in real Schur form AB09CX Hankel norm approximation based model reduction with state matrix in real Schur form AB09HX Stochastic balancing model reduction of stable systems AB09HY Cholesky factors of the controllability and observability Grammians AB09IX Accuracy enhanced balancing related model reduction AB09IY Cholesky factors of the frequency-weighted controllability and observability Grammians AB09JV State-space representation of a projection of a left weighted transfer-function matrix AB09JW State-space representation of a projection of a right weighted transfer-function matrix AB09JX Check stability/antistability of finite eigenvalues AB09KX Stable projection of V*G*W or conj(V)*G*conj(W)
AB13AX Hankel-norm of a stable system with state matrix in real Schur form AB13DX Maximum singular value of a transfer-function matrix
AG08BY Construction of a reduced system with input/output matrix Dr of full row rank, preserving the finite Smith zeros
IB01MD Upper triangular factor in QR factorization of a block-Hankel-block matrix IB01MY Upper triangular factor in fast QR factorization of a block-Hankel-block matrix IB01ND Singular value decomposition giving the system order IB01OD Estimating the system order IB01OY User's confirmation of the system order IB01PD Estimating the system matrices and covariances IB01PX Estimating the matrices B and D of a system using Kronecker products IB01PY Estimating the matrices B and D of a system exploiting the structure IB01QD Estimating the initial state and the matrices B and D of a system IB01RD Estimating the initial state of a system
MA01AD Complex square root of a complex number in real arithmetic
MA02AD Transpose of a matrix MA02BD Reversing the order of rows and/or columns of a matrix MA02CD Pertranspose of the central band of a square matrix MA02DD Pack/unpack the upper or lower triangle of a symmetric matrix MA02ED Construct a triangle of a symmetric matrix, given the other triangle MA02FD Hyperbolic plane rotation MA02GD Column interchanges on the matrix MA02HD Check if a matrix is a scalar multiple of an identity-like matrix MB01SD Rows and/or columns scaling of a matrix
MB01RU Computation of matrix expression alpha*R + beta*A*X*trans(A) (MB01RD variant) MB01RW Computation of matrix expression alpha*A*X*trans(A), X symmetric (BLAS 2) MB01RX Computing a triangle of the matrix expressions alpha*R + beta*A*B or alpha*R + beta*B*A MB01RY Computing a triangle of the matrix expressions alpha*R + beta*H*B or alpha*R + beta*B*H, with H an upper Hessenberg matrix MB01UW Computation of matrix expressions alpha*H*A or alpha*A*H, overwritting A, with H an upper Hessenberg matrix MB01VD Kronecker product of two matrices MB01XY Computation of the product U'*U or L*L', with U and L upper and lower triangular matrices (unblock algorithm) SB03OV Construction of a complex plane rotation to annihilate a real number, modifying a complex number SG03BY Computing a complex plane rotation in real arithmetic
MB02CU Bringing the first blocks of a generator in proper form (extended version of MB02CX) MB02CV Applying the MB02CU transformations on other columns / rows of the generator MB02CX Bringing the first blocks of a generator in proper form MB02CY Applying the MB02CX transformations on other columns / rows of the generator MB02NY Separation of a zero singular value of a bidiagonal submatrix MB02QY Minimum-norm least squares solution, given a rank-revealing QR factorization MB02UU Solution of linear equations using LU factorization with complete pivoting MB02UV LU factorization with complete pivoting MB02WD Solution of a positive definite linear system A*x = b, or f(A, x) = b, using conjugate gradient algorithm MB02XD Solution of a set of positive definite linear systems, A'*A*X = B, or f(A)*X = B, using Gaussian elimination MB02YD Solution of the linear system A*x = b, D*x = 0, D diagonal
MB03NY The smallest singular value of A - jwI MB03OY Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process MB03PY Matrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting) MB03QX Eigenvalues of an upper quasi-triangular matrix MB03QY Transformation to Schur canonical form of a selected 2-by-2 diagonal block of an upper quasi-triangular matrix MB03RX Reordering the diagonal blocks of a principal submatrix of a real Schur form matrix MB03RY Tentative solution of Sylvester equation -AX + XB = C (A, B in real Schur form) MB03VY Generating orthogonal matrices for reduction to periodic Hessenberg form of a product of matrices MB03WX Eigenvalues of a product of matrices, T = T_1*T_2*...*T_p, with T_1 upper quasi-triangular and T_2, ..., T_p upper triangular MB05MY Computing an orthogonal matrix reducing a matrix to real Schur form T, the eigenvalues, and the upper triangular matrix of right eigenvectors of T MB05OY Restoring a matrix after balancing transformations
MB04DY Symplectic scaling of a Hamiltonian matrix MB04IY Applying the product of elementary reflectors used for QR factorization of a matrix having a lower left zero triangle MB04NY Applying an elementary reflector to a matrix C = ( A B ), from the right, where A has one column MB04OY Applying an elementary reflector to a matrix C = ( A' B' )', from the left, where A has one row MB04OW Rank-one update of a Cholesky factorization for a 2-by-2 block matrix MB04OX Rank-one update of a Cholesky factorization MB04PY Applying an elementary reflector to a matrix from the left or right MB04TU Applying a row-permuted Givens transformation to two row vectors MB04XY Applying Householder transformations for bidiagonalization (stored in factored form) to one or two matrices, from the left MB04YW One QR or QL iteration step onto an unreduced bidiagonal submatrix of a bidiagonal matrix
MC01PY Coefficients of a real polynomial, stored in decreasing order, given its zeros
MC03NX Construction of a pencil sE-A related to a given polynomial matrix
MD03BX QR factorization with column pivoting and error vector transformation MD03BY Finding the Levenberg-Marquardt parameter
NF01AD Computing the output of a Wiener system NF01AY Computing the output of a set of neural networks NF01BD Computing the Jacobian of a Wiener system NF01BP Finding the Levenberg-Marquardt parameter NF01BQ Solution of the linear system J*x = b, D*x = 0, D diagonal NF01BR Solution of the linear system op(R)*x = b, R block upper triangular stored in a compressed form NF01BS QR factorization of a structured Jacobian matrix NF01BU Computing J'*J + c*I, for the Jacobian J given in a compressed form NF01BV Computing J'*J + c*I, for a full Jacobian J (one output variable) NF01BW Matrix-vector product x <-- (J'*J + c*I)*x, for J in a compressed form NF01BX Matrix-vector product x <-- (A'*A + c*I)*x, for a full matrix A NF01BY Computing the Jacobian of the error function for a neural network (for one output variable)
SB01BX Choosing the closest real (complex conjugate) eigenvalue(s) to a given real (complex) value SB01BY Pole placement for systems of order 1 or 2 SB01FY Inner denominator of a right-coprime factorization of an unstable system of order 1 or 2
SB02MU Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems SB02RU Constructing the 2n-by-2n Hamiltonian or symplectic matrix for linear-quadratic optimization problems (efficient and accurate version of SB02MU) SB02OY Constructing and compressing the extended Hamiltonian or symplectic matrix pairs for linear-quadratic optimization problems
SB03MV Solving a discrete-time Lyapunov equation for a 2-by-2 matrix SB03MW Solving a continuous-time Lyapunov equation for a 2-by-2 matrix SB03MX Solving a discrete-time Lyapunov equation with matrix A quasi-triangular SB03MY Solving a continuous-time Lyapunov equation with matrix A quasi-triangular SB03OT Solving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A quasi-triangular and R triangular SB03OU Solving (for Cholesky factor) stable continuous- or discrete-time Lyapunov equations, with A in real Schur form and B rectangular SB03OY Solving (for Cholesky factor) stable 2-by-2 continuous- or discrete-time Lyapunov equations, with matrix A having complex conjugate eigenvalues SB03QX Forward error bound for continuous-time Lyapunov equations SB03QY Separation and Theta norm for continuous-time Lyapunov equations SB03SX Forward error bound for discrete-time Lyapunov equations SB03SY Separation and Theta norm for discrete-time Lyapunov equations
SB03MU Solving a discrete-time Sylvester equation for an m-by-n matrix X, 1 <= m,n <= 2 SB03OR Solving quasi-triangular continuous- or discrete-time Sylvester equations, for an n-by-m matrix X, 1 <= m <= 2 SB04MR Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonal SB04MU Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the second subdiagonal SB04MW Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal SB04MY Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal SB04NV Constructing right-hand sides for a system of equations in Hessenberg form solved via SB04NX SB04NW Constructing the right-hand side for a system of equations in Hessenberg form solved via SB04NY SB04NX Solving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides SB04NY Solving a system of equations in Hessenberg form with one offdiagonal and one right-hand side SB04PX Solving a discrete-time Sylvester equation for matrices of order <= 2 SB04PY Solving a discrete-time Sylvester equation with matrices in Schur form SB04QR Solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonal SB04QU Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the third subdiagonal SB04QY Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal (discrete-time case) SB04RV Constructing right-hand sides for a system of equations in Hessenberg form solved via SB04RX SB04RW Constructing the right-hand side for a system of equations in Hessenberg form solved via SB04RY SB04RX Solving a system of equations in Hessenberg form with two consecutive offdiagonals and two right-hand sides (discrete-time case) SB04RY Solving a system of equations in Hessenberg form with one offdiagonal and one right-hand side (discrete-time case)
SB10JD Conversion of a descriptor state-space system into regular state-space form SB10LD Closed-loop system matrices for a system with robust controller SB10PD Normalization of a system for H-infinity controller design SB10QD State feedback and output injection matrices for an H-infinity (sub)optimal state controller (continuous-time) SB10RD H-infinity (sub)optimal controller matrices using state feedback and output injection matrices (continuous-time) SB10SD H2 optimal controller matrices for a normalized discrete-time system SB10TD H2 optimal controller matrices for a discrete-time system SB10UD Normalization of a system for H2 controller design SB10VD State feedback and output injection matrices for an H2 optimal state controller (continuous-time) SB10WD H2 optimal controller matrices using state feedback and output injection matrices (continuous-time) SB10YD Fitting frequency response data with a stable, minimum phase SISO system SB10ZP Transforming a SISO system into a stable and minimum phase one
SB16AY Cholesky factors of the frequency-weighted controllability and observability Grammians for controller reduction SB16CY Cholesky factors of controllability and observability Grammians of coprime factors of a state-feedback controller
SG03AX Solving a generalized discrete-time Lyapunov equation with A quasi-triangular and E upper triangular SG03AY Solving a generalized continuous-time Lyapunov equation with A quasi-triangular and E upper triangular SG03BU Solving (for Cholesky factor) stable generalized discrete-time Lyapunov equations with A quasi-triangular, and E, B upper triangular SG03BV Solving (for Cholesky factor) stable generalized continuous-time Lyapunov equations with A quasi-triangular, and E, B upper triangular SG03BX Solving (for Cholesky factor) stable generalized 2-by-2 Lyapunov equations
SG03BW Solving a generalized Sylvester equation with A quasi-triangular and E upper triangular, for X m-by-n, n = 1 or 2
TB01VD Conversion of a discrete-time system to output normal form TB01VY Conversion of the output normal form of a discrete-time system to a state-space representation TB01XD Special similarity transformation of the dual state-space system TB01YD Special similarity transformation of a state-space system
TB04BV Strictly proper part of a proper transfer function matrix TB04BW Sum of a rational matrix and a real matrix TB04BX Gain of a SISO linear system, given (A,b,c,d), its poles and zeros
TF01MX Output response of a linear discrete-time system, given a general system matrix (each output is a column of the result) TF01MY Output response of a linear discrete-time system, given the system matrices (each output is a column of the result)
TG01HX Orthogonal reduction of a descriptor system to a system with the same transfer-function matrix and without uncontrollable finite eigenvalues