Lyapunov Matrix Equation Solvers

X = lyapc(A, C)

Compute X, the symmetric or hermitian solution of the continuous Lyapunov equation

  AX + XA' + C = 0,

where A is a square real or complex matrix and C is a symmetric or hermitian matrix. A must not have two eigenvalues α and β such that α+β = 0.

The following particular cases are also adressed:

X = lyapc(α*I,C)  or  X = lyapc(α,C)

Solve the matrix equation (α+α')X + C = 0.

x = lyapc(α,γ)

Solve the equation (α+α')x + γ = 0.

Example

julia> A = [3. 4.; 5. 6.]
2×2 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

julia> C = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
 1.0  1.0
 1.0  2.0

julia> X = lyapc(A, C)
2×2 Array{Float64,2}:
  0.5  -0.5
 -0.5   0.25

julia> A*X + X*A' + C
2×2 Array{Float64,2}:
 -8.88178e-16   2.22045e-16
  2.22045e-16  -4.44089e-16

X = lyapc(A, E, C)

Compute X, the symmetric or hermitian solution of the generalized continuous Lyapunov equation

 AXE' + EXA' + C = 0,

where A and E are square real or complex matrices and C is a symmetric or hermitian matrix. The pencil A-λE must not have two eigenvalues α and β such that α+β = 0.

The following particular cases are also adressed:

X = lyapc(A,β*I,C)  or  X = lyapc(A,β,C)

Solve the matrix equation AXβ' + βXA' + C = 0.

X = lyapc(α*I,E,C)  or  X = lyapc(α,E,C)

Solve the matrix equation αXE' + EXα' + C = 0.

X = lyapc(α*I,β*I,C)  or  X = lyapc(α,β,C)

Solve the matrix equation (αβ'+α'β)X + C = 0.

x = lyapc(α,β,γ)

Solve the equation (αβ'+α'β)x + γ = 0.

Example

julia> A = [3. 4.; 5. 6.]
2×2 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

julia> E = [ 1. 2.; 0. 1.]
2×2 Array{Float64,2}:
 1.0  2.0
 0.0  1.0

julia> C = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
 1.0  1.0
 1.0  2.0

julia> X = lyapc(A, E, C)
2×2 Array{Float64,2}:
 -2.5   2.5
  2.5  -2.25

julia> A*X*E' + E*X*A' + C
2×2 Array{Float64,2}:
 -5.32907e-15  -2.66454e-15
 -4.44089e-15   0.0

lyapcs!(A,C;adj = false)

Solve the continuous Lyapunov matrix equation

            op(A)X + Xop(A)' + C = 0,

where op(A) = A if adj = false and op(A) = A' if adj = true. A is a square real matrix in a real Schur form, or a square complex matrix in a complex Schur form and C is a symmetric or hermitian matrix. A must not have two eigenvalues α and β such that α+β = 0. C contains on output the solution X.

X = lyapd(A, C)

Compute X, the symmetric or hermitian solution of the discrete Lyapunov equation

   AXA' - X + C = 0,

where A is a square real or complex matrix and C is a symmetric or hermitian matrix. A must not have two eigenvalues α and β such that αβ = 1. The following particular cases are also adressed:

X = lyapd(α*I,C)  or  X = lyapd(α,C)

Solve the matrix equation (αα'-1)X + C = 0.

x = lyapd(α,γ)

Solve the equation (αα'-1)x + γ = 0.

Example

julia> A = [3. 4.; 5. 6.]
2×2 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

julia> C = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
 1.0  1.0
 1.0  2.0

julia> X = lyapd(A, C)
2×2 Array{Float64,2}:
  0.2375  -0.2125
 -0.2125   0.1375

julia> A*X*A' - X + C
2×2 Array{Float64,2}:
 5.55112e-16  6.66134e-16
 2.22045e-16  4.44089e-16

X = lyapd(A, E, C)

Compute X, the symmetric or hermitian solution of the generalized discrete Lyapunov equation

     AXA' - EXE' + C = 0,

where A and E are square real or complex matrices and C is a symmetric or hermitian matrix. The pencil A-λE must not have two eigenvalues α and β such that αβ = 1. The following particular cases are also adressed:

X = lyapd(A,β*I,C)  or  X = lyapd(A,β,C)

Solve the matrix equation AXA' - βXβ' + C = 0.

X = lyapd(α*I,E,C)  or  X = lyapd(α,E,C)

Solve the matrix equation αXα' - EXE' + C = 0.

X = lyapd(α*I,β*I,C)  or  X = lyapd(α,β,C)

Solve the matrix equation (αα'-ββ')X + C = 0.

x = lyapd(α,β,γ)

Solve the equation (αα'-ββ')x + γ = 0.

Example

julia> A = [3. 4.; 5. 6.]
2×2 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

julia> E = [ 1. 2.; 0. -1.]
2×2 Array{Float64,2}:
 1.0   2.0
 0.0  -1.0

julia> C = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
 1.0  1.0
 1.0  2.0

julia> X = lyapd(A, E, C)
2×2 Array{Float64,2}:
  1.775  -1.225
 -1.225   0.775
 
julia> A*X*A' - E*X*E' + C
2×2 Array{Float64,2}:
 -2.22045e-16  -4.44089e-16
 -1.33227e-15   1.11022e-15

lyapds!(A, C; adj = false)

Solve the discrete Lyapunov matrix equation

            op(A)Xop(A)' - X + C = 0,

where op(A) = A if adj = false and op(A) = A' if adj = true. A is in a real or complex Schur form and C is a symmetric or hermitian matrix. A must not have two eigenvalues α and β such that αβ = 1. The computed symmetric or hermitian solution X is contained in C.

lyapds!(A, E, C; adj = false)

Solve the generalized discrete Lyapunov matrix equation

            op(A)Xop(A)' - op(E)Xop(E)' + C = 0,

where op(A) = A and op(E) = E if adj = false and op(A) = A' and op(E) = E' if adj = true. The pair (A,E) is in a generalized real or complex Schur form and C is a symmetric or hermitian matrix. The pencil A-λE must not have two eigenvalues α and β such that αβ = 1. The computed symmetric or hermitian solution X is contained in C.