The Fortran routines of the Basic Systems and Control Toolbox are aimed to perform essential system analysis and synthesis computations. The main features and options are:

• standard and generalised state space systems are covered, the main topics being:
• basic mathematical computations, including solution of linear and quadratic (symmetric) matrix equations
• system analysis and synthesis
• transfer matrix factorization
• similarity and equivalence transformations on standard and generalized state space (descriptor) systems
• availability of both, fully documented drivers and computational routines
• whenever possible, structure exploiting algorithms and dedicated linear algebra tools are used
• condition number and forward error bounds estimates can be obtained for many computations, including for Lyapunov and Riccati equations

A list of the main Fortran routines with links to the associated .html documentation is given in the following tables, where * denotes auxiliary routines:

### Mathematical routines

 MB03RD Reduction of a real Schur matrix to a block-diagonal form MB03QD Reordering of the diagonal blocks of a real Schur matrix MB03SD Eigenvalues of a square-reduced Hamiltonian matrix MB03UD Singular value decomposition of an upper triangular matrix MB04DY *Symplectic scaling of a Hamiltonian matrix MB04ZD Transforming a Hamiltonian matrix into a square-reduced form MB03VD Periodic Hessenberg form of a product of matrices MB03VY *Generating orthogonal matrices for reduction to periodic Hessenberg form of a product of matrices MB03WD Periodic Schur decomposition and eigenvalues of a product of matrices in periodic Hessenberg form MB03WX *Eigenvalues of a quasi-triangular product of matrices MB02MD Solution of Total Least-Squares problem using a singular value decomposition (SVD) approach MB02ND Solution of Total Least-Squares problem using a partial SVD approach

### Mathematical routines using HAPACK approach

 MB03TD Reordering the diagonal blocks of a matrix in (skew-)Hamiltonian Schur form MB03XD Eigenvalues of a real Hamiltonian matrix MB03XS *Eigenvalues of a real skew-Hamiltonian matrix MB03XZ Eigenvalues of a complex Hamiltonian matrix MB03XP Periodic Schur decomposition and eigenvalues of a matrix product A*B, with A upper Hessenberg and B upper triangular MB03YD Periodic QR iteration MB03ZA *Reordering a selected cluster of eigenvalues of a given matrix pair in periodic Schur form MB03ZD Stable and unstable invariant subspaces for a dichotomic Hamiltonian matrix MB04DD *Balancing a real Hamiltonian matrix MB04DZ *Balancing a complex Hamiltonian matrix MB04DI *Applying the inverse of a balancing transformation for a real Hamiltonian matrix MB04DS *Balancing a real skew-Hamiltonian matrix MB04DY *Symplectic scaling of a real Hamiltonian matrix MB04PB Paige/Van Loan form of a Hamiltonian matrix MB04QB *Applying a product of symplectic reflectors and Givens rotators to two general real matrices MB04QC *Premultiplying a real matrix with an orthogonal symplectic block reflector MB04QF *Forming the triangular block factors of a symplectic block reflector MB04TB Symplectic URV decomposition of a real 2N-by-2N matrix MB04WD *Generating an orthogonal basis spanning an isotropic subspace MB04WR *Generating orthogonal symplectic matrices defined as products of symplectic reflectors and Givens rotators SB04OW *Solving a periodic Sylvester equation with matrices in periodic Schur form UE01MD Default machine-specific parameters for (skew-)Hamiltonian computation routines

### Mathematical routines for skew-Hamiltonian/Hamiltonian pencils

 MB03BD *Finding eigenvalues of a generalized matrix product in Hessenberg-triangular form MB03BZ *Finding eigenvalues of a complex generalized matrix product in Hessenberg-triangular form MB03FZ Eigenvalues, stable right deflating subspace and companion subspace of a complex skew-Hamiltonian/Hamiltonian pencil (factored version) MB03ID *Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version) MB03IZ *Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (factored version) MB03JD *Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil MB03JZ *Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil MB03KD *Reordering the diagonal blocks of a formal matrix product in periodic Schur form using periodic QZ algorithm MB03LF Eigenvalues, stable right deflating subspace and companion subspace of a real skew-Hamiltonian/Hamiltonian pencil (factored version) MB03LD Eigenvalues and stable right deflating subspace of a real skew-Hamiltonian/Hamiltonian pencil MB03LZ Eigenvalues and stable right deflating subspace of a complex skew-Hamiltonian/Hamiltonian pencil MB04AD Eigenvalues and generalized symplectic URV decomposition of a real skew-Hamiltonian/Hamiltonian pencil in factored form MB04AZ Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil in factored form MB04BD Eigenvalues and orthogonal decomposition of a real skew-Hamiltonian/Hamiltonian pencil MB04BZ Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil MB04CD *Reducing a special real block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form (factored version) MB04ED Eigenvalues and structured Schur form of a real skew-Hamiltonian/skew-Hamiltonian pencil in factored form MB04FD Eigenvalues and structured Schur form of a real skew-Hamiltonian/skew-Hamiltonian pencil MB04HD *Reducing a special real block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form MB04QS *Matrix multiplication with a product of symplectic reflectors and Givens rotations MB04RB *Reduction of a real skew-Hamiltonian matrix to Paige/Van Loan form MB04RU *Reduction of a real skew-Hamiltonian matrix to Paige/Van Loan form (unblocked version) MB04SU *Symplectic QR decomposition of a real 2M-by-N matrix

### Mathematical routines for (block) Toeplitz matrices

 MB02CD Cholesky factor and the generator and/or the Cholesky factor of the inverse of a symmetric positive definite block Toeplitz matrix MB02CU *Bringing the first blocks of a generator in proper form (enhanced version of MB02CX) MB02CV *Applying the transformations created by the routine MB02CU on other columns or rows of the generator MB02CX *Bringing the first blocks of a generator in proper form MB02CY *Applying the transformations created by the routine MB02CX on other columns or rows of the generator MB02DD Updating the Cholesky factor, the generator, and/or the Cholesky factor of the inverse of a symmetric positive definite block Toeplitz matrix, given the information from a previous factorization and additional blocks of its first block row, or its first block column MB02ED Solving a system of linear equations, T X = B or X T = B, with a symmetric positive definite block Toeplitz matrix T MB02FD Incomplete Cholesky factor and the generator of a symmetric positive definite block Toeplitz matrix MB02GD Cholesky factor of a banded symmetric positive definite block Toeplitz matrix MB02HD Cholesky factor of the matrix T' T, with T a banded block Toeplitz matrix of full rank MB02ID Solving an over- or underdetermined linear system with a full rank block Toeplitz matrix MB02JD Full QR factorization of a block Toeplitz matrix of full rank MB02JX Low rank QR factorization with column pivoting of a block Toeplitz matrix MB02KD Computing the product C = alpha*op( T )*B + beta*C, with T a block Toeplitz matrix

### System norm and zeros routines

 AB13AD Hankel-norm of the stable projection AB13AX *Hankel-norm of a stable system with the state matrix in real Schur form AB13BD H2 or L2 norm of a system AB13DD L-infinity norm of a state space system AB13ED Complex stability radius, using bisection AB13FD Complex stability radius, using bisection and SVD AB13ID Properness test for the transfer function matrix of a descriptor system AB08ND Zeros and Kronecker structure of a system pencil AG08BD Zeros and Kronecker structure of a descriptor system pencil AB08NZ Zeros and Kronecker structure of a system pencil with complex matrices AG08BZ Zeros and Kronecker structure of a descriptor system pencil with complex matrices

### Standard systems transformation routines

 TB01ID Balancing a system matrix for a given system triple TB01IZ Balancing a system matrix for a given system triple with complex matrices TB01KD Additive spectral decomposition of a state-space system TB01LD Spectral separation of a state-space system TB01WD Reduction of the state dynamics matrix to real Schur form TB01PD Minimal, controllable or observable block Hessenberg realization

### Factorization routines

 SB08CD Left coprime factorization with inner denominator SB08DD Right coprime factorization with inner denominator SB08ED Left coprime factorization with prescribed stability degree SB08FD Right coprime factorization with prescribed stability degree SB08GD State-space representation of a left coprime factorization SB08HD State-space representation of a right coprime factorization

### Synthesis routines

 SB01BD Multiinput pole assignment using the Schur method SB01DD Eigenstructure assignment for a controllable matrix pair (A,B) in orthogonal canonical form SB06ND Minimum norm deadbeat control state feedback matrix SB02QD Condition and forward error for continuous-time Riccati equation solution SB02RD Solution of algebraic Riccati equations (generalized Schur method) SB02OD Solution of algebraic Riccati equations (refined Schur vectors method) with condition and forward error bound estimates SB02PD Solution of continuous-time algebraic Riccati equations (matrix sign function method) with condition and forward error bound estimates SG02AD Solution of algebraic Riccati equations for descriptor systems SB02SD Condition and forward error for discrete-time Riccati equation solution SB03OD Solving stable Lyapunov equations for the Cholesky factor of the solution SB03QD Condition and forward error for continuous-time Lyapunov equations SB03SD Condition and forward error for discrete-time Lyapunov equations SB03TD Solution of continuous-time Lyapunov equations, condition and forward error estimation SB03UD Solution of discrete-time Lyapunov equations, condition and forward error estimation SB04MD Solution of continuous-time Sylvester equations (Hessenberg-Schur method) SB04QD Solution of discrete-time Sylvester equations (Hessenberg-Schur method) SB04PD Solution of continuous- or discrete-time Sylvester equations (Schur method) SB04OD Solution of generalized Sylvester equations with separation estimation SG03AD Solution of generalized Lyapunov equations and separation estimation SG03BD Solution of stable generalized Lyapunov equations (Cholesky factor)

### Descriptor systems transformation routines

 TG01AD Balancing the matrices of the system pencil corresponding to a descriptor triple TG01AZ Balancing the matrices of the system pencil corresponding to a descriptor triple with complex matrices TG01CD Orthogonal reduction of a descriptor system pair (A-sE,B) to the QR-coordinate form TG01DD Orthogonal reduction of a descriptor system pair (C,A-sE) to the RQ-coordinate form TG01ED Orthogonal reduction of a descriptor system to a SVD coordinate form TG01FD Orthogonal reduction of a descriptor system to a SVD-like coordinate form TG01FZ Orthogonal reduction of a descriptor system with complex matrices to a SVD-like coordinate form TG01HD Orthogonal reduction of a descriptor system to the controllability staircase form TG01ID Orthogonal reduction of a descriptor system to the observability staircase form TG01JD Irreducible descriptor representation TG01JY Irreducible descriptor representation (blocked version)

### Benchmark routines

 BD01AD Benchmarks for state-space realizations of continuous-time dynamical systems (CTDSX) BD02AD Benchmarks for state-space realizations of discrete-time dynamical systems (DTDSX) BB01AD Benchmark examples for continuous-time Riccati equations BB02AD Benchmark examples for discrete-time Riccati equations BB03AD Benchmark examples for continuous-time Lyapunov equations (CTLEX) BB04AD Benchmark examples for discrete-time Lyapunov equations (DTLEX)

### Data analysis routines

 DE01PD Convolution or deconvolution of two real signals using Hartley transform DF01MD Sine transform or cosine transform of a real signal DK01MD Anti-aliasing window applied to a real signal

The documentation of all routines is also accessible from the SLICOT Library main index (in case of the drivers, or user-callable routines), or from the SLICOT Supporting Routines index (in case of the auxiliary routines, marked with * in the tables above). The SLICOT Supporting Routines index is also accessible from the main Library index.

This email address is being protected from spambots. You need JavaScript enabled to view it. March 10, 2005; Updated: December 2, 2012