Package Modelica.Math.Matrices
Library of functions operating on matrices

Information

Library content

This library provides functions operating on matrices. Below, the functions are ordered according to categories and a typical call of the respective function is shown. Most functions are solely an interface to the external LAPACK library.

Note: A' is a short hand notation of transpose(A):

Basic Information

toString(A) - returns the string representation of matrix A.
isEqual(M1, M2) - returns true if matrices M1 and M2 have the same size and the same elements.

Linear Equations

solve(A,b) - returns solution x of the linear equation A*x=b (where b is a vector, and A is a square matrix that must be regular).
solve2(A,B) - returns solution X of the linear equation A*X=B (where B is a matrix, and A is a square matrix that must be regular)
leastSquares(A,b) - returns solution x of the linear equation A*x=b in a least squares sense (where b is a vector and A may be non-square and may be rank deficient)
leastSquares2(A,B) - returns solution X of the linear equation A*X=B in a least squares sense (where B is a matrix and A may be non-square and may be rank deficient)
equalityLeastSquares(A,a,B,b) - returns solution x of a linear equality constrained least squares problem: min|A*x-a|^2 subject to B*x=b
(LU,p,info) = LU(A) - returns the LU decomposition with row pivoting of a rectangular matrix A.
LU_solve(LU,p,b) - returns solution x of the linear equation L*U*x[p]=b with a b vector and an LU decomposition from "LU(..)".
LU_solve2(LU,p,B) - returns solution X of the linear equation L*U*X[p,:]=B with a B matrix and an LU decomposition from "LU(..)".

Matrix Factorizations

(eval,evec) = eigenValues(A) - returns eigen values "eval" and eigen vectors "evec" for a real, nonsymmetric matrix A in a Real representation.
eigenValueMatrix(eval) - returns real valued block diagonal matrix of the eigenvalues "eval" of matrix A.
(sigma,U,VT) = singularValues(A) - returns singular values "sigma" and left and right singular vectors U and VT of a rectangular matrix A.
(Q,R,p) = QR(A) - returns the QR decomposition with column pivoting of a rectangular matrix A such that Q*R = A[:,p].
(H,U) = hessenberg(A) - returns the upper Hessenberg form H and the orthogonal transformation matrix U of a square matrix A such that H = U'*A*U.
realSchur(A) - returns the real Schur form of a square matrix A.
cholesky(A) - returns the cholesky factor H of a real symmetric positive definite matrix A so that A = H'*H.
(D,Aimproved) = balance(A) - returns an improved form Aimproved of a square matrix A that has a smaller condition as A, with Aimproved = inv(diagonal(D))*A*diagonal(D).

Matrix Properties

trace(A) - returns the trace of square matrix A, i.e., the sum of the diagonal elements.
det(A) - returns the determinant of square matrix A (using LU decomposition; try to avoid det(..))
inv(A) - returns the inverse of square matrix A (try to avoid, use instead "solve2(..) with B=identity(..))
rank(A) - returns the rank of square matrix A (computed with singular value decomposition)
conditionNumber(A) - returns the condition number norm(A)*norm(inv(A)) of a square matrix A in the range 1..∞.
rcond(A) - returns the reciprocal condition number 1/conditionNumber(A) of a square matrix A in the range 0..1.
norm(A) - returns the 1-, 2-, or infinity-norm of matrix A.
frobeniusNorm(A) - returns the Frobenius norm of matrix A.
nullSpace(A) - returns the null space of matrix A.

Matrix Exponentials

exp(A) - returns the exponential e^A of a matrix A by adaptive Taylor series expansion with scaling and balancing
(phi, gamma) = integralExp(A,B) - returns the exponential phi=e^A and the integral gamma=integral(exp(A*t)*dt)*B as needed for a discretized system with zero order hold.
(phi, gamma, gamma1) = integralExpT(A,B) - returns the exponential phi=e^A, the integral gamma=integral(exp(A*t)*dt)*B, and the time-weighted integral gamma1 = integral((T-t)*exp(A*t)*dt)*B as needed for a discretized system with first order hold.

Matrix Equations

continuousLyapunov(A,C) - returns solution X of the continuous-time Lyapunov equation X*A + A'*X = C
continuousSylvester(A,B,C) - returns solution X of the continuous-time Sylvester equation A*X + X*B = C
continuousRiccati(A,B,R,Q) - returns solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0
discreteLyapunov(A,C) - returns solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C
discreteSylvester(A,B,C) - returns solution X of the discrete-time Sylvester equation A*X*B + sgn*X = C
discreteRiccati(A,B,R,Q) - returns solution X of the discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

Matrix Manipulation

sort(M) - returns the sorted rows or columns of matrix M in ascending or descending order.
flipLeftRight(M) - returns matrix M so that the columns of M are flipped in left/right direction.
flipUpDown(M) - returns matrix M so that the rows of M are flipped in up/down direction.

Package Contents

Name	Description
`balance`	Return a balanced form of matrix A to improve the condition of A
`balanceABC`	Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation
`cholesky`	Return the Cholesky factorization of a symmetric positive definite matrix
`conditionNumber`	Return the condition number norm(A)*norm(inv(A)) of a matrix A
`continuousLyapunov`	Return solution X of the continuous-time Lyapunov equation XA + A'X = C
`continuousRiccati`	Return solution X of the continuous-time algebraic Riccati equation A'X + XA - XBinv(R)B'X + Q = 0 (care)
`continuousSylvester`	Return solution X of the continuous-time Sylvester equation AX + XB = C
`det`	Return determinant of a matrix (computed by LU decomposition; try to avoid det(..))
`discreteLyapunov`	Return solution X of the discrete-time Lyapunov equation A'XA + sgn*X = C
`discreteRiccati`	Return solution of discrete-time algebraic Riccati equation A'XA - X - A'XBinv(R + B'XB)B'XA + Q = 0 (dare)
`discreteSylvester`	Return solution of the discrete-time Sylvester equation AXB + sgn*X = C
`eigenValueMatrix`	Return real valued block diagonal matrix J of eigenvalues of matrix A (A=VJVinv)
`eigenValues`	Return eigenvalues for a real, nonsymmetric matrix in a Real representation
`equalityLeastSquares`	Solve a linear equality constrained least squares problem
`Examples` …	Examples demonstrating the usage of the Math.Matrices functions
`exp`	Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing
`flipLeftRight`	Flip the columns of a matrix in left/right direction
`flipUpDown`	Flip the rows of a matrix in up/down direction
`frobeniusNorm`	Return the Frobenius norm of a matrix
`hessenberg`	Return upper Hessenberg form of a matrix
`integralExp`	Return the exponential and the integral of the exponential of a matrix
`integralExpT`	Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix
`inv`	Return inverse of a matrix (try to avoid inv(..))
`isEqual`	Compare whether two Real matrices are identical
`LAPACK` …	Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)
`leastSquares`	Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
`leastSquares2`	Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)
`LU`	LU decomposition of square or rectangular matrix
`LU_solve`	Solve real system of linear equations PLU*x=b with a b vector and an LU decomposition (from LU(..))
`LU_solve2`	Solve real system of linear equations PLU*X=B with a B matrix and an LU decomposition (from LU(..))
`norm`	Return the p-norm of a matrix
`nullSpace`	Return the orthonormal nullspace of a matrix
`QR`	Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)
`rank`	Return rank of a rectangular matrix (computed with singular values)
`rcond`	Return the reciprocal condition number of a matrix
`realSchur`	Return the real Schur form (rsf) S of a square matrix A, A=QZSQZ'
`singularValues`	Return singular values and left and right singular vectors
`solve`	Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)
`solve2`	Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)
`sort`	Sort the rows or columns of a matrix in ascending or descending order
`toString`	Convert a matrix into its string representation
`trace`	Return the trace of matrix A, i.e., the sum of the diagonal elements
`Utilities` …	Utility functions that should not be directly utilized by the user

Function Modelica.Math.Matrices.toString
Convert a matrix into its string representation

Information

Syntax

Matrices.toString(A);
Matrices.toString(A, name="", significantDigits=6);

Description

The function call "Matrices.toString(A)" returns the string representation of matrix A. With the optional arguments "name" and "significantDigits", a name and the number of the digits are defined. The default values of name and significantDigits are "" and 6 respectively. If name=="" then the prefix "<name> =" is left out.

Example

  A = [2.12, -4.34; -2.56, -1.67];

  toString(A);
  // = "
  //      2.12   -4.34
  //     -2.56   -1.67";

  toString(A,"A",1);
  // = "A =
  //         2     -4
  //        -3     -2"

Inputs

Type	Name	Description
`Real`	`M[:,:]`	Real matrix
`String`	`name`	Independent variable name used for printing
`Integer`	`significantDigits`	Number of significant digits that are shown

Outputs

Type	Name	Description
`String`	`s`	String expression of matrix M

Function Modelica.Math.Matrices.isEqual
Compare whether two Real matrices are identical

Information

Syntax

Matrices.isEqual(M1, M2);
Matrices.isEqual(M1, M2, eps=0);

Description

The function call "Matrices.isEqual(M1, M2)" returns true, if the two Real matrices M1 and M2 have the same dimensions and the same elements. Otherwise the function returns false. Two elements e1 and e2 of the two matrices are checked on equality by the test "abs(e1-e2) ≤ eps", where "eps" can be provided as third argument of the function. Default is "eps = 0".

Example

  Real A1[2,2] = [1,2; 3,4];
  Real A2[3,2] = [1,2; 3,4; 5,6];
  Real A3[2,2] = [1,2, 3,4.0001];
  Boolean result;
algorithm
  result := Matrices.isEqual(M1,M2);     // = false
  result := Matrices.isEqual(M1,M3);     // = false
  result := Matrices.isEqual(M1,M1);     // = true
  result := Matrices.isEqual(M1,M3,0.1); // = true

Inputs

Type	Name	Description
`Real`	`M1[:,:]`	First matrix
`Real`	`M2[:,:]`	Second matrix (may have different size as M1)
`Real`	`eps`	Two elements e1 and e2 of the two matrices are identical if abs(e1-e2) <= eps

Outputs

Type	Name	Description
`Boolean`	`result`	= true, if matrices have the same size and the same elements

Function Modelica.Math.Matrices.solve
Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)

Information

Syntax

Matrices.solve(A,b);

Description

This function call returns the solution x of the linear system of equations

A*x = b

If a unique solution x does not exist (since A is singular), an assertion is triggered. If this is not desired, use instead Matrices.leastSquares and inquire the singularity of the solution with the return argument rank (a unique solution is computed if rank = size(A,1)).

Note, the solution is computed with the LAPACK function "dgesv", i.e., by Gaussian elimination with partial pivoting.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b[3] = {10,22,12};
  Real x[3];
algorithm
  x := Matrices.solve(A,b);  // x = {3,2,1}

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Matrix A of A*x = b
`Real`	`b[size(A, 1)]`	Vector b of A*x = b

Outputs

Type	Name	Description
`Real`	`x[size(b, 1)]`	Vector x such that A*x = b

Function Modelica.Math.Matrices.solve2
Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)

Information

Syntax

Matrices.solve2(A,b);

Description

This function call returns the solution X of the linear system of equations

A*X = B

If a unique solution X does not exist (since A is singular), an assertion is triggered. If this is not desired, use instead Matrices.leastSquares2 and inquire the singularity of the solution with the return argument rank (a unique solution is computed if rank = size(A,1)).

Note, the solution is computed with the LAPACK function "dgesv", i.e., by Gaussian elimination with partial pivoting.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real B[3,2] = [10, 20;
                 22, 44;
                 12, 24];
  Real X[3,2];
algorithm
  X := Matrices.solve2(A, B);  /* X = [3, 6;
                                       2, 4;
                                       1, 2] */

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Matrix A of A*X = B
`Real`	`B[size(A, 1),:]`	Matrix B of A*X = B

Outputs

Type	Name	Description
`Real`	`X[size(B, 1),size(B, 2)]`	Matrix X such that A*X = B

Function Modelica.Math.Matrices.leastSquares
Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Information

Syntax

x = Matrices.leastSquares(A,b);

Description

Returns a solution of equation A*x = b in a least square sense (A may be rank deficient):

  minimize | A*x - b |

Several different cases can be distinguished (note, rank is an output argument of this function):

size(A,1) = size(A,2)

A solution is returned for a regular, as well as a singular matrix A:

rank = size(A,1):
A is regular and the returned solution x fulfills the equation A*x = b uniquely.
rank < size(A,1):
A is singular and no unique solution for equation A*x = b exists.
- If an infinite number of solutions exists, the one is selected that fulfills the equation and at the same time has the minimum norm |x| for all solution vectors that fulfill the equation.
- If no solution exists, x is selected such that |A*x - b| is as small as possible (but A*x - b is not zero).

size(A,1) > size(A,2):

The equation A*x = b has no unique solution. The solution x is selected such that |A*x - b| is as small as possible. If rank = size(A,2), this minimum norm solution is unique. If rank < size(A,2), there are an infinite number of solutions leading to the same minimum value of |A*x - b|. From these infinite number of solutions, the one with the minimum norm |x| is selected. This gives a unique solution that minimizes both |A*x - b| and |x|.

size(A,1) < size(A,2):

rank = size(A,1):
There are an infinite number of solutions that fulfill the equation A*x = b. From this infinite number, the unique solution is selected that minimizes |x|.
rank < size(A,1):
There is either no solution of equation A*x = b, or there are again an infinite number of solutions. The unique solution x is returned that minimizes both |A*x - b| and |x|.

Note, the solution is computed with the LAPACK function "dgelsy", i.e., QR or LQ factorization of A with column pivoting.

Algorithmic details

The function first computes a QR factorization with column pivoting:

      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]

with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:

     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]

The minimum-norm solution is then

     x = P * Z' [ inv(T11)*Q1'*b ]
                [        0       ]

where Q1 consists of the first "rank" columns of Q.

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix A
`Real`	`b[size(A, 1)]`	Vector b
`Real`	`rcond`	Reciprocal condition number to estimate the rank of A

Outputs

Type	Name	Description
`Real`	`x[size(A, 2)]`	Vector x such that min\|Ax-b\|^2 if size(A,1) >= size(A,2) or min\|x\|^2 and Ax=b, if size(A,1) < size(A,2)
`Integer`	`rank`	Rank of A

Function Modelica.Math.Matrices.leastSquares2
Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Information

Syntax

X = Matrices.leastSquares2(A,B);

Description

Returns a solution of equation A*X = B in a least square sense (A may be rank deficient):

  minimize | A*X - B |

Several different cases can be distinguished (note, rank is an output argument of this function):

size(A,1) = size(A,2)

A solution is returned for a regular, as well as a singular matrix A:

rank = size(A,1):
A is regular and the returned solution X fulfills the equation A*X = B uniquely.
rank < size(A,1):
A is singular and no unique solution for equation A*X = B exists.
- If an infinite number of solutions exists, the one is selected that fulfills the equation and at the same time has the minimum norm |x| for all solution vectors that fulfill the equation.
- If no solution exists, X is selected such that |A*X - B| is as small as possible (but A*X - B is not zero).

size(A,1) > size(A,2):

The equation A*X = B has no unique solution. The solution X is selected such that |A*X - B| is as small as possible. If rank = size(A,2), this minimum norm solution is unique. If rank < size(A,2), there are an infinite number of solutions leading to the same minimum value of |A*X - B|. From these infinite number of solutions, the one with the minimum norm |X| is selected. This gives a unique solution that minimizes both |A*X - B| and |X|.

size(A,1) < size(A,2):

rank = size(A,1):
There are an infinite number of solutions that fulfill the equation A*X = B. From this infinite number, the unique solution is selected that minimizes |X|.
rank < size(A,1):
There is either no solution of equation A*X = B, or there are again an infinite number of solutions. The unique solution X is returned that minimizes both |A*X - B| and |X|.

Note, the solution is computed with the LAPACK function "dgelsy", i.e., QR or LQ factorization of A with column pivoting.

Algorithmic details

The function first computes a QR factorization with column pivoting:

      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]

with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:

     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]

The minimum-norm solution is then

     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]

where Q1 consists of the first "rank" columns of Q.

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix A
`Real`	`B[size(A, 1),:]`	Matrix B
`Real`	`rcond`	Reciprocal condition number to estimate rank of A

Outputs

Type	Name	Description
`Real`	`X[size(A, 2),size(B, 2)]`	Matrix X such that min\|AX-B\|^2 if size(A,1) >= size(A,2) or min\|X\|^2 and AX=B, if size(A,1) < size(A,2)
`Integer`	`rank`	Rank of A

Function Modelica.Math.Matrices.equalityLeastSquares
Solve a linear equality constrained least squares problem

Information

Syntax

x = Matrices.equalityLeastSquares(A,a,B,b);

Description

This function returns the solution x of the linear equality-constrained least squares problem:

min|A*x - a|^2 over x, subject to B*x = b

It is required that the dimensions of A and B fulfill the following relationship:

size(B,1) ≤ size(A,2) ≤ size(A,1) + size(B,1)

Note, the solution is computed with the LAPACK function "dgglse" using the generalized RQ factorization under the assumptions that B has full row rank (= size(B,1)) and the matrix [A;B] has full column rank (= size(A,2)). In this case, the problem has a unique solution.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Minimize \|A*x - a\|^2
`Real`	`a[size(A, 1)]`
`Real`	`B[:,size(A, 2)]`	subject to B*x=b
`Real`	`b[size(B, 1)]`

Outputs

Type	Name	Description
`Real`	`x[size(A, 2)]`	solution vector

Function Modelica.Math.Matrices.LU
LU decomposition of square or rectangular matrix

Information

Syntax

(LU, pivots)       = Matrices.LU(A);
(LU, pivots, info) = Matrices.LU(A);

Description

This function call returns the LU decomposition of a "Real[m,n]" matrix A, i.e.,

P*L*U = A

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). Matrices L and U are stored in the returned matrix LU (the diagonal of L is not stored). With the companion function Matrices.LU_solve, this decomposition can be used to solve linear systems (P*L*U)*x = b with different right hand side vectors b. If a linear system of equations with just one right hand side vector b shall be solved, it is more convenient to just use the function Matrices.solve.

The optional third (Integer) output argument has the following meaning:

info = 0:	successful exit
info > 0:	if info = i, U[i,i] is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

The LU factorization is computed with the LAPACK function "dgetrf", i.e., by Gaussian elimination using partial pivoting with row interchanges. Vector "pivots" are the pivot indices, i.e., for 1 ≤ i ≤ min(m,n), row i of matrix A was interchanged with row pivots[i].

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b1[3] = {10,22,12};
  Real b2[3] = { 7,13,10};
  Real    LU[3,3];
  Integer pivots[3];
  Real    x1[3];
  Real    x2[3];
algorithm
  (LU, pivots) := Matrices.LU(A);
  x1 := Matrices.LU_solve(LU, pivots, b1);  // x1 = {3,2,1}
  x2 := Matrices.LU_solve(LU, pivots, b2);  // x2 = {1,0,2}

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Square or rectangular matrix

Outputs

Type	Name	Description
`Real`	`LU[size(A, 1),size(A, 2)]`	L,U factors (used with LU_solve(..))
`Integer`	`pivots[min(size(A, 1), size(A, 2))]`	pivot indices (used with LU_solve(..))
`Integer`	`info`	Information

Function Modelica.Math.Matrices.LU_solve
Solve real system of linear equations PLU*x=b with a b vector and an LU decomposition (from LU(..))

Information

Syntax

Matrices.LU_solve(LU, pivots, b);

Description

This function call returns the solution x of the linear systems of equations

P*L*U*x = b;

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). The matrices of this decomposition are computed with function Matrices.LU that returns arguments LU and pivots used as input arguments of Matrices.LU_solve. With Matrices.LU and Matrices.LU_solve it is possible to efficiently solve linear systems with different right hand side vectors. If a linear system of equations with just one right hand side vector shall be solved, it is more convenient to just use the function Matrices.solve.

If a unique solution x does not exist (since the LU decomposition is singular), an exception is raised.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real b1[3] = {10,22,12};
  Real b2[3] = { 7,13,10};
  Real    LU[3,3];
  Integer pivots[3];
  Real    x1[3];
  Real    x2[3];
algorithm
  (LU, pivots) := Matrices.LU(A);
  x1 := Matrices.LU_solve(LU, pivots, b1);  // x1 = {3,2,1}
  x2 := Matrices.LU_solve(LU, pivots, b2);  // x2 = {1,0,2}

Inputs

Type	Name	Description
`Real`	`LU[:,size(LU, 1)]`	L,U factors of Matrices.LU(..) for a square matrix
`Integer`	`pivots[size(LU, 1)]`	Pivots indices of Matrices.LU(..)
`Real`	`b[size(LU, 1)]`	Right hand side vector of PLU*x=b

Outputs

Type	Name	Description
`Real`	`x[size(b, 1)]`	Solution vector such that PLU*x = b

Function Modelica.Math.Matrices.LU_solve2
Solve real system of linear equations PLU*X=B with a B matrix and an LU decomposition (from LU(..))

Information

Syntax

Matrices.LU_solve2(LU, pivots, B);

Description

This function call returns the solution X of the linear systems of equations

P*L*U*X = B;

where P is a permutation matrix (implicitly defined by vector pivots), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). The matrices of this decomposition are computed with function Matrices.LU that returns arguments LU and pivots used as input arguments of Matrices.LU_solve2. With Matrices.LU and Matrices.LU_solve2 it is possible to efficiently solve linear systems with different right hand side matrices. If a linear system of equations with just one right hand side matrix shall be solved, it is more convenient to just use the function Matrices.solve2.

If a unique solution X does not exist (since the LU decomposition is singular), an exception is raised.

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real B1[3] = [10, 20;
                22, 44;
                12, 24];
  Real B2[3] = [ 7, 14;
                13, 26;
                10, 20];
  Real    LU[3,3];
  Integer pivots[3];
  Real    X1[3,2];
  Real    X2[3,2];
algorithm
  (LU, pivots) := Matrices.LU(A);
  X1 := Matrices.LU_solve2(LU, pivots, B1);  /* X1 = [3, 6;
                                                      2, 4;
                                                      1, 2] */
  X2 := Matrices.LU_solve2(LU, pivots, B2);  /* X2 = [1, 2;
                                                      0, 0;
                                                      2, 4] */

Inputs

Type	Name	Description
`Real`	`LU[:,size(LU, 1)]`	L,U factors of Matrices.LU(..) for a square matrix
`Integer`	`pivots[size(LU, 1)]`	Pivots indices of Matrices.LU(..)
`Real`	`B[size(LU, 1),:]`	Right hand side matrix of PLU*X=B

Outputs

Type	Name	Description
`Real`	`X[size(B, 1),size(B, 2)]`	Solution matrix such that PLU*X = B

Function Modelica.Math.Matrices.eigenValues
Return eigenvalues for a real, nonsymmetric matrix in a Real representation

Information

Syntax

                eigenvalues = Matrices.eigenValues(A);
(eigenvalues, eigenvectors) = Matrices.eigenValues(A);

Description

This function call returns the eigenvalues and optionally the (right) eigenvectors of a square matrix A. The first column of "eigenvalues" contains the real and the second column contains the imaginary part of the eigenvalues. If the i-th eigenvalue has no imaginary part, then eigenvectors[:,i] is the corresponding real eigenvector. If the i-th eigenvalue has an imaginary part, then eigenvalues[i+1,:] is the conjugate complex eigenvalue and eigenvectors[:,i] is the real and eigenvectors[:,i+1] is the imaginary part of the eigenvector of the i-th eigenvalue. With function Matrices.eigenValueMatrix, a real block diagonal matrix is constructed from the eigenvalues such that

A = eigenvectors * eigenValueMatrix(eigenvalues) * inv(eigenvectors)

provided the eigenvector matrix "eigenvectors" can be inverted (an inversion is possible, if all eigenvalues are different; in some cases, an inversion is also possible if some eigenvalues are the same).

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real eval;
algorithm
  eval := Matrices.eigenValues(A);  // eval = [-0.618, 0;
                                    //          8.0  , 0;
                                    //          1.618, 0];

i.e., matrix A has the 3 real eigenvalues -0.618, 8, 1.618.

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Matrix

Outputs

Type	Name	Description
`Real`	`eigenvalues[size(A, 1),2]`	Eigenvalues of matrix A (Re: first column, Im: second column)
`Real`	`eigenvectors[size(A, 1),size(A, 2)]`	Real-valued eigenvector matrix

Function Modelica.Math.Matrices.eigenValueMatrix
Return real valued block diagonal matrix J of eigenvalues of matrix A (A=VJVinv)

Information

Syntax

Matrices.eigenValueMatrix(eigenvalues);

Description

The function call returns a block diagonal matrix J from the two-column matrix eigenvalues (computed by function Matrices.eigenValues). Matrix eigenvalues must have the real part of the eigenvalues in the first column and the imaginary part in the second column. If an eigenvalue i has a vanishing imaginary part, then J[i,i] = eigenvalues[i,1], i.e., the diagonal element of J is the real eigenvalue. Otherwise, eigenvalue i and conjugate complex eigenvalue i+1 are used to construct a 2 by 2 diagonal block of J:

  J[i  , i]   := eigenvalues[i,1];
  J[i  , i+1] := eigenvalues[i,2];
  J[i+1, i]   := eigenvalues[i+1,2];
  J[i+1, i+1] := eigenvalues[i+1,1];

Inputs

Type	Name	Description
`Real`	`eigenValues[:,2]`	Eigen values from function eigenValues(..) (Re: first column, Im: second column)

Outputs

Type	Name	Description
`Real`	`J[size(eigenValues, 1),size(eigenValues, 1)]`	Real valued block diagonal matrix with eigen values (Re: 1x1 block, Im: 2x2 block)

Function Modelica.Math.Matrices.singularValues
Return singular values and left and right singular vectors

Information

Syntax

         sigma = Matrices.singularValues(A);
(sigma, U, VT) = Matrices.singularValues(A);

Description

This function computes the singular values and optionally the singular vectors of matrix A. Basically the singular value decomposition of A is computed, i.e.,

A = U S V^T
  = U*Sigma*VT

where U and V are orthogonal matrices (UU^T=I, VV^T=I). S = [diagonal(s_i), zeros(n,m-n)], if n=size(A,1) ≤ m=size(A,2)) or [diagonal(s_i); zeros(n-m,m)], if n > m=size(A,2)). S has the same size as matrix A with nonnegative diagonal elements in decreasing order and with all other elements zero (s₁ is the largest element). The function returns the singular values s_i in vector sigma and the orthogonal matrices in matrices U and VT.

Example

  A = [1, 2,  3,  4;
       3, 4,  5, -2;
      -1, 2, -3,  5];
  (sigma, U, VT) = singularValues(A);
  results in:
     sigma = {8.33, 6.94, 2.31};
  i.e.
     Sigma = [8.33,    0,    0, 0;
                 0, 6.94,    0, 0;
                 0,    0, 2.31, 0]

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix

Outputs

Type	Name	Description
`Real`	`sigma[min(size(A, 1), size(A, 2))]`	Singular values
`Real`	`U[size(A, 1),size(A, 1)]`	Left orthogonal matrix
`Real`	`VT[size(A, 2),size(A, 2)]`	Transposed right orthogonal matrix

Function Modelica.Math.Matrices.QR
Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)

Information

Syntax

(Q,R,p) = Matrices.QR(A);

Description

This function returns the QR decomposition of a rectangular matrix A (the number of columns of A must be less than or equal to the number of rows):

Q*R = A[:,p]

where Q is a rectangular matrix that has orthonormal columns and has the same size as A (Q^TQ=I), R is a square, upper triangular matrix and p is a permutation vector. Matrix R has the following important properties:

The absolute value of a diagonal element of R is the largest value in this row, i.e., abs(R[i,i]) ≥ abs(R[i,j]).
The diagonal elements of R are sorted according to size, such that the largest absolute value is abs(R[1,1]) and abs(R[i,i]) ≥ abs(R[j,j]) with i < j.

This means that if abs(R[i,i]) ≤ ε then abs(R[j,k]) ≤ ε for j ≥ i, i.e., the i-th row up to the last row of R have small elements and can be treated as being zero. This allows to, e.g., estimate the row-rank of R (which is the same row-rank as A). Furthermore, R can be partitioned in two parts

   A[:,p] = Q * [R₁, R₂;
                 0,  0]

where R₁ is a regular, upper triangular matrix.

Note, the solution is computed with the LAPACK functions "dgeqpf" and "dorgqr", i.e., by Householder transformations with column pivoting. If Q is not needed, the function may be called as: (,R,p) = QR(A).

Example

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real R[3,3];
algorithm
  (,R) := Matrices.QR(A);  // R = [-7.07.., -4.24.., -3.67..;
                                    0     , -1.73.., -0.23..;
                                    0     ,  0     ,  0.65..];

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Rectangular matrix with size(A,1) >= size(A,2)
`Boolean`	`pivoting`	True if column pivoting is performed. True is default

Outputs

Type	Name	Description
`Real`	`Q[size(A, 1),size(A, 2)]`	Rectangular matrix with orthonormal columns such that Q*R=A[:,p]
`Real`	`R[size(A, 2),size(A, 2)]`	Square upper triangular matrix
`Integer`	`p[size(A, 2)]`	Column permutation vector

Function Modelica.Math.Matrices.hessenberg
Return upper Hessenberg form of a matrix

Information

Syntax

         H = Matrices.hessenberg(A);
    (H, U) = Matrices.hessenberg(A);

Description

Function hessenberg computes the Hessenberg matrix H of matrix A as well as the orthogonal transformation matrix U that holds H = U'*A*U. The Hessenberg form of a matrix is computed by repeated Householder similarity transformation. The elementary reflectors and the corresponding scalar factors are provided by function "Utilities.toUpperHessenberg()". The transformation matrix U is then computed by LAPACK.dorghr.

Example

 A  = [1, 2,  3;
       6, 5,  4;
       1, 0,  0];

 (H, U) = hessenberg(A);

  results in:

 H = [1.0,  -2.466,  2.630;
     -6.083, 5.514, -3.081;
      0.0,   0.919, -0.514]

 U = [1.0,    0.0,      0.0;
      0.0,   -0.9864,  -0.1644;
      0.0,   -0.1644,   0.9864]

  and therefore,

 U*H*transpose(U) = [1.0, 2.0, 3.0;
                     6.0, 5.0, 4.0;
                     1.0, 0.0, 0.0]

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A

Outputs

Type	Name	Description
`Real`	`H[size(A, 1),size(A, 2)]`	Hessenberg form of A
`Real`	`U[size(A, 1),size(A, 2)]`	Transformation matrix

Function Modelica.Math.Matrices.realSchur
Return the real Schur form (rsf) S of a square matrix A, A=QZSQZ'

Information

Syntax

                            S = Matrices.realSchur(A);
(S, QZ, alphaReal, alphaImag) = Matrices.realSchur(A);

Description

Function realSchur calculates the real Schur form of a real square matrix A, i.e.

 A = QZ*S*transpose(QZ)

with the real nxn matrices S and QZ. S is a block upper triangular matrix with 1x1 and 2x2 blocks in the diagonal. QZ is an orthogonal matrix. The 1x1 blocks contains the real eigenvalues of A. The 2x2 blocks [s11, s12; s21, s11] represents the conjugated complex pairs of eigenvalues, whereas the real parts of the eigenvalues are the elements of the diagonal (s11). The imaginary parts are the positive and negative square roots of the product of the two elements s12 and s21 (imag = +-sqrt(s12*s21)).

The calculation in lapack.dgees is performed stepwise, i.e., using the internal methods of balancing and scaling of dgees.

Example

   Real A[3,3] = [1, 2, 3; 4, 5, 6; 7, 8, 9];
   Real T[3,3];
   Real Z[3,3];
   Real alphaReal[3];
   Real alphaImag[3];

algorithm
  (T, Z, alphaReal, alphaImag):=Modelica.Math.Matrices.realSchur(A);
//   T = [16.12, 4.9,   1.59E-015;
//        0,    -1.12, -1.12E-015;
//        0,     0,    -1.30E-015]
//   Z = [-0.23,  -0.88,   0.41;
//        -0.52,  -0.24,  -0.82;
//        -0.82,   0.4,    0.41]
//alphaReal = {16.12, -1.12, -1.32E-015}
//alphaImag = {0, 0, 0}

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix

Outputs

Type	Name	Description
`Real`	`S[size(A, 1),size(A, 2)]`	Real Schur form of A
`Real`	`QZ[size(A, 1),size(A, 2)]`	Schur vector Matrix
`Real`	`alphaReal[size(A, 1)]`	Real part of eigenvalue=alphaReal+i*alphaImag
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of eigenvalue=alphaReal+i*alphaImag

Function Modelica.Math.Matrices.cholesky
Return the Cholesky factorization of a symmetric positive definite matrix

Information

Syntax

         H = Matrices.cholesky(A);
         H = Matrices.cholesky(A, upper=true);

Description

Function cholesky computes the Cholesky factorization of a real symmetric positive definite matrix A. The optional Boolean input "upper" specifies whether the upper or the lower triangular matrix is returned, i.e.

 A = H'*H   if upper is true (H is upper triangular)
 A = H*H'   if upper is false (H is lower triangular)

The computation is performed by LAPACK.dpotrf.

Example

  A  = [1, 0,  0;
        6, 5,  0;
        1, -2,  2];
  S = A*transpose(A);

  H = Matrices.cholesky(S);

  results in:

  H = [1.0,  6.0,  1.0;
       0.0,  5.0, -2.0;
       0.0,  0.0,  2.0]

  with

  transpose(H)*H = [1.0,  6.0,   1;
                    6.0, 61.0,  -4.0;
                    1.0, -4.0,   9.0] //=S

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Symmetric positive definite matrix
`Boolean`	`upper`	True if the right cholesky factor (upper triangle) should be returned

Outputs

Type	Name	Description
`Real`	`H[size(A, 1),size(A, 2)]`	Cholesky factor U (upper=true) or L (upper=false) for A = U'U or A = LL'

Function Modelica.Math.Matrices.balance
Return a balanced form of matrix A to improve the condition of A

Information

Syntax

(D,B) = Matrices.balance(A);

Description

This function returns a vector D, such that B=inv(diagonal(D))*A*diagonal(D) has a better condition as matrix A, i.e., conditionNumber(B) ≤ conditionNumber(A). The elements of D are multiples of 2 which means that this function does not introduce round-off errors. Balancing attempts to make the norm of each row of B equal to the norm of the respective column.

Balancing is used to minimize roundoff errors induced through large matrix calculations like Taylor-series approximation or computation of eigenvalues.

Example

       - A = [1, 10,  1000; 0.01,  0,  10; 0.005,  0.01,  10]
       - Matrices.norm(A, 1);
         = 1020.0
       - (T,B)=Matrices.balance(A)
       - T
         = {256, 16, 0.5}
       - B
         =  [1,     0.625,   1.953125;
             0.16,  0,       0.3125;
             2.56,  0.32,   10.0]
       - Matrices.norm(B, 1);
         = 12.265625

The Algorithm is taken from

H. D. Joos, G. Grübel:: RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which based on the balance function from EISPACK.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`D[size(A, 1)]`	diagonal(D)=T is transformation matrix, such that B = inv(T)AT has smaller condition as A
`Real`	`B[size(A, 1),size(A, 1)]`	Balanced matrix (= inv(diagonal(D))Adiagonal(D) )

Function Modelica.Math.Matrices.balanceABC
Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation

Information

Syntax

(scale,As,Bs,Cs) = Matrices.balanceABC(A,B,C);
(scale,As,Bs)    = Matrices.balanceABC(A,B);
(scale,As,,Cs)   = Matrices.balanceABC(A,C=C);

Description

This function returns a vector scale, such that with T=diagonal(scale) system matrix S_scale

             |inv(T)*A*T, inv(T)*B|
   S_scale = |                    |
             |       C*T,     0   |

has a better condition as system matrix S

       |A, B|
   S = |    |
       |C, 0|

that is, conditionNumber(S_scale) ≤ conditionNumber(S). The elements of vector scale are multiples of 2 which means that this function does not introduce round-off errors.

Balancing a linear dynamic system in state space form

  der(x) = A*x + B*u
      y  = C*x + D*u

means to find a state transformation x_new = T*x = diagonal(scale)*x so that the transformed system is better suited for numerical algorithms.

Example

import Modelica.Math.Matrices;

A = [1, -10,  1000; 0.01,  0,  10; 0.005,  -0.01,  10];
B = [100, 10; 1,0; -0.003, 1];
C = [-0.5, 1, 100];

(scale, As, Bs, Cs) := Matrices.balanceABC(A,B,C);
T    = diagonal(scale);
Diff = [Matrices.inv(T)*A*T, Matrices.inv(T)*B;
        C*T, zeros(1,2)] - [As, Bs; Cs, zeros(1,2)];
err  = Matrices.norm(Diff);

-> Results in:
scale = {16, 1, 0.0625}
norm(A)  = 1000.15, norm(B)  = 100.504, norm(C)  = 100.006
norm(As) = 10.8738, norm(Bs) = 16.0136, norm(Cs) = 10.2011
err = 0

The algorithm is taken from

H. D. Joos, G. Grübel:: RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which is based on the balance function from EISPACK.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	System matrix A
`Real`	`B[size(A, 1),:]`	System matrix B (need not be present)
`Real`	`C[:,size(A, 1)]`	System matrix C (need not be present)

Outputs

Type	Name	Description
`Real`	`scale[size(A, 1)]`	diagonal(scale)=T is such that [inv(T)AT, inv(T)B; CT, 0] has smaller condition as [A,B;C,0]
`Real`	`As[size(A, 1),size(A, 1)]`	Balanced matrix A (= inv(T)AT )
`Real`	`Bs[size(A, 1),size(B, 2)]`	Balanced matrix B (= inv(T)*B )
`Real`	`Cs[size(C, 1),size(A, 1)]`	Balanced matrix C (= C*T )

Function Modelica.Math.Matrices.trace
Return the trace of matrix A, i.e., the sum of the diagonal elements

Information

Syntax

  r = Matrices.trace(A);

Description

This function computes the trace, i.e., the sum of the elements in the diagonal of matrix A.

Example

  A = [1, 3;
       2, 1];
  r = trace(A);

  results in:

  r = 2.0

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A

Outputs

Type	Name	Description
`Real`	`result`	Trace of A

Function Modelica.Math.Matrices.det
Return determinant of a matrix (computed by LU decomposition; try to avoid det(..))

Information

Syntax

result = Matrices.det(A);

Description

This function returns the determinant "result" of matrix A computed by a LU decomposition with row pivoting. For details about determinants, see http://en.wikipedia.org/wiki/Determinant. Usually, this function should never be used, because there are nearly always better numerical algorithms as by computing the determinant. Examples:

Use Matrices.rank to compute whether det(A) = 0 (i.e., Matrices.rank(A) < size(A,1)).
Use Matrices.solve to solve the linear equation A*x = b, instead of using determinants to compute the solution.

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`result`	Determinant of matrix A

Function Modelica.Math.Matrices.inv
Return inverse of a matrix (try to avoid inv(..))

Information

Syntax

invA = Matrices.inv(A);

Description

This function returns the inverse of matrix A, i.e., A*inv(A) = identity(size(A,1)) computed by a LU decomposition with row pivoting. Usually, this function should not be used, because there are nearly always better numerical algorithms as by computing directly the inverse. Example:

Use x = Matrices.solve(A,b) to solve the linear equation A*x = b, instead of computing the solution by x = inv(A)*b, because this is much more efficient and much more reliable.

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`invA[size(A, 1),size(A, 2)]`	Inverse of matrix A

Function Modelica.Math.Matrices.rank
Return rank of a rectangular matrix (computed with singular values)

Information

Syntax

result = Matrices.rank(A);
result = Matrices.rank(A,eps=0);

Description

This function returns the rank of a square or rectangular matrix A computed by singular value decomposition. For details about the rank of a matrix, see http://en.wikipedia.org/wiki/Matrix_rank. To be more precise:

rank(A) returns the number of singular values of A that are larger than max(size(A))*norm(A)*Modelica.Constants.eps.
rank(A, eps) returns the number of singular values of A that are larger than "eps".

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix
`Real`	`eps`	If eps > 0, the singular values are checked against eps; otherwise eps=max(size(A))norm(A)Modelica.Constants.eps is used

Outputs

Type	Name	Description
`Integer`	`result`	Rank of matrix A

Function Modelica.Math.Matrices.conditionNumber
Return the condition number norm(A)*norm(inv(A)) of a matrix A

Information

Syntax

r = Matrices.conditionNumber(A);

Description

This function calculates the condition number (norm(A) * norm(inv(A))) of a general real matrix A, in either the 1-norm, 2-norm or the infinity-norm. In the case of 2-norm the result is the ratio of the largest to the smallest singular value of A. For more details, see http://en.wikipedia.org/wiki/Condition_number.

Example

  A = [1, 2;
       2, 1];
  r = conditionNumber(A);

  results in:

  r = 3.0

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Input matrix
`Real`	`p`	Type of p-norm (only allowed: 1, 2 or Modelica.Constants.inf)

Outputs

Type	Name	Description
`Real`	`result`	Condition number of matrix A

Function Modelica.Math.Matrices.rcond
Return the reciprocal condition number of a matrix

Information

Syntax

r = Matrices.rcond(A);

Description

This function estimates the reciprocal of the condition number (norm(A) * norm(inv(A))) of a general real matrix A, in either the 1-norm or the infinity-norm, using the LAPACK function DGECON. If rcond(A) is near 1.0, A is well conditioned and A is ill conditioned if rcond(A) is near zero.

Example

  A = [1, 2;
       2, 1];
  r = rcond(A);

  results in:

  r = 0.3333

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square real matrix
`Boolean`	`inf`	Is true if infinity norm is used and false for 1-norm

Outputs

Type	Name	Description
`Real`	`rcond`	Reciprocal condition number of A
`Integer`	`info`	Information

Function Modelica.Math.Matrices.norm
Return the p-norm of a matrix

Information

Syntax

Matrices.norm(A);
Matrices.norm(A, p=2);

Description

The function call "Matrices.norm(A)" returns the 2-norm of matrix A, i.e., the largest singular value of A.
The function call "Matrices.norm(A, p)" returns the p-norm of matrix A. The only allowed values for p are

"p=1": the largest column sum of A
"p=2": the largest singular value of A
"p=Modelica.Constants.inf": the largest row sum of A

Note, for any matrices A1, A2 the following inequality holds:

Matrices.norm(A1+A2,p) ≤ Matrices.norm(A1,p) + Matrices.norm(A2,p)

Note, for any matrix A and vector v the following inequality holds:

Vectors.norm(A*v,p) ≤ Matrices.norm(A,p)*Vectors.norm(A,p)

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Input matrix
`Real`	`p`	Type of p-norm (only allowed: 1, 2 or Modelica.Constants.inf)

Outputs

Type	Name	Description
`Real`	`result`	p-norm of matrix A

Function Modelica.Math.Matrices.frobeniusNorm
Return the Frobenius norm of a matrix

Information

Syntax

  r = Matrices.frobeniusNorm(A);

Description

This function computes the Frobenius norm of a general real matrix A, i.e., the square root of the sum of the squares of all elements.

Example

  A = [1, 2;
       2, 1];
  r = frobeniusNorm(A);

  results in:

  r = 3.162;

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Input matrix

Outputs

Type	Name	Description
`Real`	`result`	Frobenius norm of matrix A

Function Modelica.Math.Matrices.nullSpace
Return the orthonormal nullspace of a matrix

Information

Syntax

           Z = Matrices.nullspace(A);
(Z, nullity) = Matrices.nullspace(A);

Description

This function calculates an orthonormal basis Z=[z_1, z_2, ...] of the nullspace of a matrix A, i.e., A*z_i=0.

The nullspace is obtained by SVD method. That is, matrix A is decomposed into the matrices S, U, V:

 A = U*S*transpose(V)

with the orthonormal matrices U and V and the matrix S with

 S = [S1, 0]
 S1 = [diag(s); 0]

and the singular values s={s1, s2, ..., sr} of A and r=rank(A). Note, that S has the same size as A. Since U and V are orthonormal we may write

 transpose(U)*A*V = [S1, 0].

Matrix S1 obviously has full column rank and therefore, the left n-r rows (n is the number of columns of A or S) of matrix V span a nullspace of A.

The nullity of matrix A is the dimension of the nullspace of A. In view of the above, it becomes clear that nullity holds

 nullity = n - r

with

 n = number of columns of matrix A
 r = rank(A)

Example

  A = [1, 2,  3, 1;
       3, 4,  5, 2;
      -1, 2, -3, 3];
  (Z, nullity) = nullspace(A);

  results in:

  Z=[0.1715;
    -0.686;
     0.1715;
     0.686]

  nullity = 1

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Input matrix

Outputs

Type	Name	Description
`Real`	`Z[size(A, 2),:]`	Orthonormal nullspace of matrix A
`Integer`	`nullity`	Nullity, i.e., the dimension of the nullspace

Function Modelica.Math.Matrices.exp
Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing

Information

Syntax

phi = Matrices.exp(A);
phi = Matrices.exp(A,T=1);

Description

This function computes the exponential e^AT of matrix A, i.e.

                            (AT)^2   (AT)^3
      Φ = e^(AT) = I + AT + ------ + ------ + ....
                              2!       3!

where e=2.71828..., A is an n x n matrix with real elements and T is a real number, e.g., the sampling time. A may be singular. With the exponential of a matrix it is, e.g., possible to compute the solution of a linear system of differential equations

    der(x) = A*x   ->   x(t0 + T) = e^(AT)*x(t0)

Algorithmic details:

The algorithm is taken from

H. D. Joos, G. Grübel:: RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

The following steps are performed to calculate the exponential of A:

Matrix A is balanced
(= is transformed with a diagonal matrix D, such that inv(D)*A*D has a smaller condition as A).
The scalar T is divided by a multiple of 2 such that norm( inv(D)*A*D*T/2^k ) < 0.5. Note, that (1) and (2) are implemented such that no round-off errors are introduced.
The matrix from (2) is approximated by explicitly performing the Taylor series expansion with a variable number of terms. Truncation occurs if a new term does no longer contribute to the value of Φ from the previous iteration.
The resulting matrix is transformed back, by reverting the steps of (2) and (1).

In several sources it is not recommended to use Taylor series expansion to calculate the exponential of a matrix, such as in 'C.B. Moler and C.F. Van Loan: Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20, pp. 801-836, 1979' or in the documentation of m-file expm2 in MATLAB version 6 (http://www.mathworks.com) where it is stated that 'As a practical numerical method, this is often slow and inaccurate'. These statements are valid for a direct implementation of the Taylor series expansion, but not for the implementation variant used in this function.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`T`

Outputs

Type	Name	Description
`Real`	`phi[size(A, 1),size(A, 1)]`	= exp(A*T)

Function Modelica.Math.Matrices.integralExp
Return the exponential and the integral of the exponential of a matrix

Information

Syntax

(phi,gamma) = Matrices.integralExp(A,B);
(phi,gamma) = Matrices.integralExp(A,B,T=1);

Description

This function computes the exponential phi = e^(AT) of matrix A and the integral gamma = integral(phi*dt)*B.

The function uses a Taylor series expansion with Balancing and scaling/squaring to approximate the integral Ψ of the matrix exponential Φ=e^(AT):

                                 AT^2   A^2 * T^3          A^k * T^(k+1)
        Ψ = int(e^(As))ds = IT + ---- + --------- + ... + --------------
                                  2!        3!                (k+1)!

Φ is calculated through Φ = I + A*Ψ, so A may be singular. Γ is simply Ψ*B.

The algorithm runs in the following steps:

Balancing
Scaling
Taylor series expansion
Re-scaling
Re-Balancing

Balancing put the bad condition of a square matrix A into a diagonal transformation matrix D. This reduce the effort of following calculations. Afterwards the result have to be re-balanced by transformation D*Atransf *inv(D).
Scaling halfen T k-times, until the norm of A*T is less than 0.5. This guarantees minimum rounding errors in the following series expansion. The re-scaling based on the equation exp(A*2T) = exp(AT)^2. The needed re-scaling formula for psi thus becomes:

         Φ = Φ'*Φ'
   I + A*Ψ = I + 2A*Ψ' + A^2*Ψ'^2
         Ψ = A*Ψ'^2 + 2*Ψ'

where psi' is the scaled result from the series expansion while psi is the re-scaled matrix.

The function is normally used to discretize a state-space system as the zero-order-hold equivalent:

      x(k+1) = Φ*x(k) + Γ*u(k)
        y(k) = C*x(k) + D*u(k)

The zero-order-hold sampling, also known as step-invariant method, gives exact values of the state variables, under the assumption that the control signal u is constant between the sampling instants. Zero-order-hold sampling is described in

K. J. Åström, B. Wittenmark:: Computer Controlled Systems - Theory and Design
Third Edition, p. 32

Syntax:
      (phi,gamma) = Matrices.expIntegral(A,B,T)
                       A,phi: [n,n] square matrices
                     B,gamma: [n,m] input matrix
                           T: scalar, e.g., sampling time

The Algorithm to calculate psi is taken from

H. D. Joos, G. Grübel:: RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),:]`
`Real`	`T`

Outputs

Type	Name	Description
`Real`	`phi[size(A, 1),size(A, 1)]`	= exp(A*T)
`Real`	`gamma[size(A, 1),size(B, 2)]`	= integral(phi)*B

Function Modelica.Math.Matrices.integralExpT
Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix

Information

(phi,gamma,gamma1) = Matrices.integralExp(A,B);
(phi,gamma,gamma1) = Matrices.integralExp(A,B,T=1);

Description

This function computes the exponential phi = e^(AT) of matrix A and the integral gamma = integral(phi*dt)*B and the integral integral((T-t)*exp(A*t)*dt)*B, where A is a square (n,n) matrix and B, gamma, and gamma1 are (n,m) matrices.

The function calculates the matrices phi,gamma,gamma1 through the equation:

                                 [ A B 0 ]
[phi gamma gamma1] = [I 0 0]*exp([ 0 0 I ]*T)
                                 [ 0 0 0 ]

The matrices define the discretized first-order-hold equivalent of a state-space system:

      x(k+1) = phi*x(k) + gamma*u(k) + gamma1/T*(u(k+1) - u(k))

The first-order-hold sampling, also known as ramp-invariant method, gives more smooth control signals as the ZOH equivalent. First-order-hold sampling is, e.g., described in

K. J. Åström, B. Wittenmark:: Computer Controlled Systems - Theory and Design
Third Edition, p. 256

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),:]`
`Real`	`T`

Outputs

Type	Name	Description
`Real`	`phi[size(A, 1),size(A, 1)]`	= exp(A*T)
`Real`	`gamma[size(A, 1),size(B, 2)]`	= integral(phi)*B
`Real`	`gamma1[size(A, 1),size(B, 2)]`	= integral((T-t)exp(At))*B

Function Modelica.Math.Matrices.continuousLyapunov
Return solution X of the continuous-time Lyapunov equation XA + A'X = C

Information

Syntax

         X = Matrices.continuousLyapunov(A, C);
         X = Matrices.continuousLyapunov(A, C, ATisSchur, eps);

Description

This function computes the solution X of the continuous-time Lyapunov equation

 X*A + A'*X = C

using the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

 Y*R' + R*Y = D

with R=U'*A'*U is the real Schur form of A' and D=U'*C*U and Y=U'*X*U are the corresponding transformations of C and X. This problem is solved sequentially for the 1x1 or 2x2 Schur blocks by exploiting the block triangular form of R. Finally the solution of the original problem is recovered as X=U*Y*U'.
The Boolean input "ATisSchur" indicates to omit the transformation to Schur in the case that A' has already Schur form.

References

  [1] Bartels, R.H. and Stewart G.W.
      Algorithm 432: Solution of the matrix equation AX + XB = C.
      Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

  A = [1, 2,  3,  4;
       3, 4,  5, -2;
      -1, 2, -3, -5;
       0, 2,  0,  6];

  C =  [-2, 3, 1, 0;
        -6, 8, 0, 1;
         2, 3, 4, 5;
        0, -2, 0, 0];

  X = continuousLyapunov(A, C);

  results in:

  X = [1.633, -0.761,  0.575, -0.656;
      -1.158,  1.216,  0.047,  0.343;
      -1.066, -0.052, -0.916,  1.61;
      -2.473,  0.717, -0.986,  1.48]

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A in XA + A'X = C
`Real`	`C[size(A, 1),size(A, 2)]`	Square matrix C in XA + A'X = C
`Boolean`	`ATisSchur`	True if transpose(A) has already real Schur form
`Real`	`eps`	Tolerance eps

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(A, 2)]`	Solution X of the Lyapunov equation XA + A'X = C

Function Modelica.Math.Matrices.continuousSylvester
Return solution X of the continuous-time Sylvester equation AX + XB = C

Information

Syntax

         X = Matrices.continuousSylvester(A, B, C);
         X = Matrices.continuousSylvester(A, B, C, AisSchur, BisSchur);

Description

Function continuousSylvester computes the solution X of the continuous-time Sylvester equation

 A*X + X*B = C.

using the Schur method for Sylvester equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

 S*Y + Y*T = D.

with S=U'*A*U is the real Schur of A, T=V'*T*V is the real Schur form of B and D=U'*C*V and Y=U*X*V' are the corresponding transformations of C and X. This problem is solved sequentially by exploiting the block triangular form of S and T. Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisSchur" and "BisSchur" indicate to omit one or both of the transformation to Schur in the case that A and/or B have already Schur form.

The function applies LAPACK-routine DTRSYL. See LAPACK.dtrsyl for more information.

References

  [1] Bartels, R.H. and Stewart G.W.
      Algorithm 432: Solution of the matrix equation AX + XB = C.
      Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

  A = [17.0,   24.0,   1.0,   8.0,   15.0 ;
       23.0,    5.0,   7.0,  14.0,   16.0 ;
        0.0,    6.0,  13.0,  20.0,   22.0;
        0.0,    0.0,  19.0,  21.0,    3.0 ;
        0.0,    0.0,   0.0,   2.0,    9.0];

  B =  [8.0, 1.0, 6.0;
        0.0, 5.0, 7.0;
        0.0, 9.0, 2.0];

  C = [62.0,  -12.0, 26.0;
       59.0,  -10.0, 31.0;
       70.0,  -6.0,   9.0;
       35.0,  31.0,  -7.0;
       36.0, -15.0,   7.0];

  X = continuousSylvester(A, B, C);

  results in:

  X = [0.0,  0.0,  1.0;
       1.0,  0.0,  0.0;
       0.0,  1.0,  0.0;
       1.0,  1.0, -1.0;
       2.0, -2.0,  1.0];

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Square matrix A
`Real`	`B[:,:]`	Square matrix B
`Real`	`C[size(A, 1),size(B, 2)]`	Matrix C
`Boolean`	`AisSchur`	True if A has already real Schur form
`Boolean`	`BisSchur`	True if B has already real Schur form

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(B, 2)]`	Solution of the continuous Sylvester equation

Function Modelica.Math.Matrices.continuousRiccati
Return solution X of the continuous-time algebraic Riccati equation A'X + XA - XBinv(R)B'X + Q = 0 (care)

Information

Syntax

                                X = Matrices.continuousRiccati(A, B, R, Q);
        (X, alphaReal, alphaImag) = Matrices.continuousRiccati(A, B, R, Q, true);

Description

Function continuousRiccati computes the solution X of the continuous-time algebraic Riccati equation

 A'*X + X*A - X*G*X + Q = 0

with G = B*inv(R)*B' using the Schur vector approach proposed by Laub [1].

It is assumed that Q is symmetric and positive semidefinite and R is symmetric, nonsingular and positive definite, (A,B) is stabilizable and (A,Q) is detectable.

These assumptions are not checked in this function !!

The assumptions guarantee that the Hamiltonian matrix

H = [A, -G; -Q, -A']

has no pure imaginary eigenvalue and can be put to an ordered real Schur form

U'*H*U = S = [S11, S12; 0, S22]

with orthogonal similarity transformation U. S is ordered in such a way, that S11 contains the n stable eigenvalues of the closed loop system with system matrix A - B*inv(R)*B'*X. If U is partitioned to

U = [U11, U12; U21, U22]

with dimensions according to S, the solution X is calculated by

X*U11 = U21.

With optional input refinement=true a subsequent iterative refinement based on Newton's method with exact line search is applied. See continuousRiccatiIterative for more information.

References

  [1] Laub, A.J.
      A Schur Method for Solving Algebraic Riccati equations.
      IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.

Example

  A = [0.0, 1.0;
       0.0, 0.0];

  B = [0.0;
       1.0];

  R = [1];

  Q = [1.0, 0.0;
       0.0, 2.0];

X = continuousRiccati(A, B, R, Q);

  results in:

X = [2.0, 1.0;
     1.0, 2.0];

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A in CARE
`Real`	`B[size(A, 1),:]`	Matrix B in CARE
`Real`	`R[size(B, 2),size(B, 2)]`	Matrix R in CARE
`Real`	`Q[size(A, 1),size(A, 1)]`	Matrix Q in CARE
`Boolean`	`refine`	True for subsequent refinement

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(A, 2)]`	stabilizing solution of CARE
`Real`	`alphaReal[2 * size(A, 1)]`	Real parts of eigenvalue=alphaReal+i*alphaImag
`Real`	`alphaImag[2 * size(A, 1)]`	Imaginary parts of eigenvalue=alphaReal+i*alphaImag

Function Modelica.Math.Matrices.discreteLyapunov
Return solution X of the discrete-time Lyapunov equation A'XA + sgn*X = C

Information

Syntax

         X = Matrices.discreteLyapunov(A, C);
         X = Matrices.discreteLyapunov(A, C, ATisSchur, sgn, eps);

Description

This function computes the solution X of the discrete-time Lyapunov equation

 A'*X*A + sgn*X = C

where sgn=1 or sgn =-1. For sgn = -1, the discrete Lyapunov equation is a special case of the Stein equation:

 A*X*B - X + Q = 0.

The algorithm uses the Schur method for Lyapunov equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

 R*Y*R' + sgn*Y = D.

with R=U'*A'*U is the real Schur form of A' and D=U'*C*U and Y=U'*X*U are the corresponding transformations of C and X. This problem is solved sequentially by exploiting the block triangular form of R. Finally the solution of the original problem is recovered as X=U*Y*U'.
The Boolean input "ATisSchur" indicates to omit the transformation to Schur in the case that A' has already Schur form.

References

  [1] Bartels, R.H. and Stewart G.W.
      Algorithm 432: Solution of the matrix equation AX + XB = C.
      Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

  A = [1, 2,  3,  4;
       3, 4,  5, -2;
      -1, 2, -3, -5;
       0, 2,  0,  6];

  C =  [-2,  3, 1, 0;
        -6,  8, 0, 1;
         2,  3, 4, 5;
         0, -2, 0, 0];

  X = discreteLyapunov(A, C, sgn=-1);

  results in:

  X  = [7.5735,   -3.1426,  2.7205, -2.5958;
       -2.6105,    1.2384, -0.9232,  0.9632;
        6.6090,   -2.6775,  2.6415, -2.6928;
       -0.3572,    0.2298,  0.0533, -0.27410];

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A in A'XA + sgn*X = C
`Real`	`C[size(A, 1),size(A, 2)]`	Square matrix C in A'XA + sgn*X = C
`Boolean`	`ATisSchur`	True if transpose(A) has already real Schur form
`Integer`	`sgn`	Specifies the sign in A'XA + sgn*X = C
`Real`	`eps`	Tolerance eps

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(A, 2)]`	Solution X of the Lyapunov equation A'XA + sgn*X = C

Function Modelica.Math.Matrices.discreteSylvester
Return solution of the discrete-time Sylvester equation AXB + sgn*X = C

Information

Syntax

         X = Matrices.discreteSylvester(A, B, C);
         X = Matrices.discreteSylvester(A, B, C, AisHess, BTisSchur, sgn, eps);

Description

Function discreteSylvester computes the solution X of the discrete-time Sylvester equation

 A*X*B + sgn*X = C.

where sgn = 1 or sgn = -1. The algorithm applies the Hessenberg-Schur method proposed by Golub et al [1]. For sgn = -1, the discrete Sylvester equation is also known as Stein equation:

 A*X*B - X + Q = 0.

In a nutshell, the problem is reduced to the corresponding problem

 H*Y*S' + sgn*Y = F.

with H=U'*A*U is the Hessenberg form of A and S=V'*B'*V is the real Schur form of B', F=U'*C*V and Y=U*X*V' are appropriate transformations of C and X. This problem is solved sequentially by exploiting the specific forms of S and H. Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisHess" and "BTisSchur" indicate to omit one or both of the transformation to Hessenberg form or Schur form respectively in the case that A and/or B have already Hessenberg form or Schur respectively.

References

  [1] Golub, G.H., Nash, S. and Van Loan, C.F.
      A Hessenberg-Schur method for the problem AX + XB = C.
      IEEE Transaction on Automatic Control, AC-24, no. 6, pp. 909-913, 1979.

Example

  A = [1.0,   2.0,   3.0;
       6.0,   7.0,   8.0;
       9.0,   2.0,   3.0];

  B = [7.0,   2.0,   3.0;
       2.0,   1.0,   2.0;
       3.0,   4.0,   1.0];

  C = [271.0,   135.0,   147.0;
       923.0,   494.0,   482.0;
       578.0,   383.0,   287.0];

  X = discreteSylvester(A, B, C);

  results in:
  X = [2.0,   3.0,   6.0;
       4.0,   7.0,   1.0;
       5.0,   3.0,   2.0];

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A in AXB + sgn*X = C
`Real`	`B[:,size(B, 1)]`	Square matrix B in AXB + sgn*X = C
`Real`	`C[size(A, 2),size(B, 1)]`	Rectangular matrix C in AXB + sgn*X = C
`Boolean`	`AisHess`	True if A has already Hessenberg form
`Boolean`	`BTisSchur`	True if B' has already real Schur form
`Integer`	`sgn`	Specifies the sign in AXB + sgn*X = C
`Real`	`eps`	Tolerance

Outputs

Type	Name	Description
`Real`	`X[size(A, 2),size(B, 1)]`	solution of the discrete Sylvester equation AXB + sgn*X = C

Function Modelica.Math.Matrices.discreteRiccati
Return solution of discrete-time algebraic Riccati equation A'XA - X - A'XBinv(R + B'XB)B'XA + Q = 0 (dare)

Information

Syntax

                                 X = Matrices.discreteRiccati(A, B, R, Q);
         (X, alphaReal, alphaImag) = Matrices.discreteRiccati(A, B, R, Q, true);

Description

Function discreteRiccati computes the solution X of the discrete-time algebraic Riccati equation

 A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0

using the Schur vector approach proposed by Laub [1].

It is assumed that Q is symmetric and positive semidefinite and R is symmetric, nonsingular and positive definite, (A,B) is stabilizable and (A,Q) is detectable. Using this method, A has also to be invertible.

These assumptions are not checked in this function !!!

The assumptions guarantee that the Hamiltonian matrix.

H = [A + G*T*Q, -G*T; -T*Q, T]

with

     -T
T = A

and

       -1
G = B*R *B'

has no eigenvalue on the unit circle and can be put to an ordered real Schur form

U'*H*U = S = [S11, S12; 0, S22]

with orthogonal similarity transformation U. S is ordered in such a way, that S11 contains the n stable eigenvalues of the closed loop system with system matrix

                  -1
A - B*(R + B'*X*B)  *B'*X*A

If U is partitioned to

U = [U11, U12; U21, U22]

according to S, the solution X can be calculated by

X*U11 = U21.

References

  [1] Laub, A.J.
      A Schur Method for Solving Algebraic Riccati equations.
      IEEE Trans. Auto. Contr., AC-24, pp. 913-921, 1979.

Example

 A  = [4.0    3.0]
      -4.5,  -3.5];

 B  = [ 1.0;
       -1.0];

 R = [1.0];

 Q = [9.0, 6.0;
      6.0, 4.0]

X = discreteRiccati(A, B, R, Q);

  results in:

X = [14.5623, 9.7082;
      9.7082, 6.4721];

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix A in DARE
`Real`	`B[size(A, 1),:]`	Matrix B in DARE
`Real`	`R[size(B, 2),size(B, 2)]`	Matrix R in DARE
`Real`	`Q[size(A, 1),size(A, 1)]`	Matrix Q in DARE
`Boolean`	`refine`	True for subsequent refinement

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(A, 2)]`	orthogonal matrix of the Schur vectors associated to ordered rsf
`Real`	`alphaReal[2 * size(A, 1)]`	Real part of eigenvalue=alphaReal+i*alphaImag
`Real`	`alphaImag[2 * size(A, 1)]`	Imaginary part of eigenvalue=alphaReal+i*alphaImag

Function Modelica.Math.Matrices.sort
Sort the rows or columns of a matrix in ascending or descending order

Information

Syntax

           sorted_M = Matrices.sort(M);
(sorted_M, indices) = Matrices.sort(M, sortRows=true, ascending=true);

Description

Function sort(..) sorts the rows of a Real matrix M in ascending order and returns the result in sorted_M. If the optional argument "sortRows" is false, the columns of the matrix are sorted. If the optional argument "ascending" is false, the rows or columns are sorted in descending order. In the optional second output argument, the indices of the sorted rows or columns with respect to the original matrix are given, such that

   sorted_M = if sortedRow then M[indices,:] else M[:,indices];

Example

  (M2, i2) := Matrices.sort([2, 1,  0;
                             2, 0, -1]);
       -> M2 = [2, 0, -1;
                2, 1, 0 ];
          i2 = {2,1};

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`M[:,:]`	Matrix to be sorted
`Boolean`	`sortRows`	= true if rows are sorted, otherwise columns
`Boolean`	`ascending`	= true if ascending order, otherwise descending order

Outputs

Type	Name	Description
`Real`	`sorted_M[size(M, 1),size(M, 2)]`	Sorted matrix
`Integer`	`indices[if sortRows then size(M, 1) else size(M, 2)]`	sorted_M = if sortRows then M[indices,:] else M[:,indices]

Function Modelica.Math.Matrices.flipLeftRight
Flip the columns of a matrix in left/right direction

Information

Syntax

         A_flr = Matrices.flipLeftRight(A);

Description

Function flipLeftRight computes from matrix A a matrix A_flr with flipped columns, i.e., A_flr[:,i]=A[:,n-i+1], i=1,..., n.

Example

  A = [1, 2,  3;
       3, 4,  5;
      -1, 2, -3];

  A_flr = flipLeftRight(A);

  results in:

  A_flr = [3, 2,  1;
           5, 4,  3;
          -3, 2, -1]

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix to be flipped

Outputs

Type	Name	Description
`Real`	`Aflip[size(A, 1),size(A, 2)]`	Flipped matrix

Function Modelica.Math.Matrices.flipUpDown
Flip the rows of a matrix in up/down direction

Information

Syntax

         A_fud = Matrices.flipUpDown(A);

Description

Function flipUpDown computes from matrix A a matrix A_fud with flipped rows, i.e., A_fud[i,:]=A[n-i+1,:], i=1,..., n.

Example

  A = [1, 2,  3;
       3, 4,  5;
      -1, 2, -3];

  A_fud = flipUpDown(A);

  results in:

  A_fud  = [-1, 2, -3;
             3, 4,  5;
             1, 2,  3]

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Matrix to be flipped

Outputs

Type	Name	Description
`Real`	`Aflip[size(A, 1),size(A, 2)]`	Flipped matrix

Package Modelica.​Math.​MatricesLibrary of functions operating on matrices

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Function Modelica.​Math.​Matrices.​toStringConvert a matrix into its string representation

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Function Modelica.​Math.​Matrices.​isEqualCompare whether two Real matrices are identical

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Function Modelica.​Math.​Matrices.​solveSolve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)

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Function Modelica.​Math.​Matrices.​solve2Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)

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Function Modelica.​Math.​Matrices.​leastSquaresSolve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

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Function Modelica.​Math.​Matrices.​leastSquares2Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

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Function Modelica.​Math.​Matrices.​equalityLeastSquaresSolve a linear equality constrained least squares problem

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Function Modelica.​Math.​Matrices.​LULU decomposition of square or rectangular matrix

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Function Modelica.​Math.​Matrices.​LU_solveSolve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..))

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Function Modelica.​Math.​Matrices.​LU_solve2Solve real system of linear equations P*L*U*X=B with a B matrix and an LU decomposition (from LU(..))

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Package Modelica.Math.Matrices
Library of functions operating on matrices

Function Modelica.Math.Matrices.toString
Convert a matrix into its string representation

Function Modelica.Math.Matrices.isEqual
Compare whether two Real matrices are identical

Function Modelica.Math.Matrices.solve
Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting)

Function Modelica.Math.Matrices.solve2
Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting)

Function Modelica.Math.Matrices.leastSquares
Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Function Modelica.Math.Matrices.leastSquares2
Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient)

Function Modelica.Math.Matrices.equalityLeastSquares
Solve a linear equality constrained least squares problem

Function Modelica.Math.Matrices.LU
LU decomposition of square or rectangular matrix

Function Modelica.Math.Matrices.LU_solve
Solve real system of linear equations PLU*x=b with a b vector and an LU decomposition (from LU(..))

Function Modelica.Math.Matrices.LU_solve2
Solve real system of linear equations PLU*X=B with a B matrix and an LU decomposition (from LU(..))

Function Modelica.Math.Matrices.eigenValues
Return eigenvalues for a real, nonsymmetric matrix in a Real representation

Function Modelica.Math.Matrices.eigenValueMatrix
Return real valued block diagonal matrix J of eigenvalues of matrix A (A=VJVinv)

Function Modelica.Math.Matrices.singularValues
Return singular values and left and right singular vectors

Function Modelica.Math.Matrices.QR
Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R)

Function Modelica.Math.Matrices.hessenberg
Return upper Hessenberg form of a matrix

Function Modelica.Math.Matrices.realSchur
Return the real Schur form (rsf) S of a square matrix A, A=QZSQZ'

Function Modelica.Math.Matrices.cholesky
Return the Cholesky factorization of a symmetric positive definite matrix

Function Modelica.Math.Matrices.balance
Return a balanced form of matrix A to improve the condition of A

Function Modelica.Math.Matrices.balanceABC
Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation