Method for providing error information relating to inconsistencies in a system of differential equations
View Patent ↗The method and the computer-related products provide for error information relating to inconsistencies in a system of differential equations that describes a technical system or a technical process.
1. A method for simulating an electrical network, whereby the inconsistencies in a system of differential equations of the electrical network are detected by the following steps:
providing a computer system including a memory, an arithmetic unit, an input device and an output device;
providing a system of equations describing the electrical network, the equations having the form f (t, x (t), {dot over (x)} (t), . . . , x (k) (t), p )= 0 , including:
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wherein x (t) and derivatives {dot over (x)} (t), . . . , x (k) thereof in each case have m number of_elements, and p is a parameter vector that can occur in the system of equations; and
executing a test method for providing error information relating to inconsistencies in the system of equations on the computer system, the test method having the following steps 1 to 3:
step 1: setting up a dependence matrix A with m number of columns and n number of rows, and setting an element A to
A (i, j)≠0 when an i th row of f (t, x (t), {dot over (x)} (t), . . . , x (k) (t),p) defined with f i (t, x (t), {dot over (x)} (t), . . ., x (k) (t)) is a function of
a) a j th element of x expressed as x j (t); or
b) one of the derivatives of the j th element of x defined as x j (s) (t);
and otherwise setting the element A (i, j)=0;
step 2: determining a set of row ranks each having the numbers of those rows of the dependence matrix A that are mutually dependent and determining a set of column ranks of the dependence matrix A each having the numbers of those columns of the dependence matrix A that are mutually dependent if such row ranks and/or column ranks are present; and
step 3: outputting at the output device error information containing the numbers contained in each row rank determined in step 2 and in each column rank determined in step 2;
wherein the row rank includes a set C 2 of natural numbers i, where 1≦i≦n, of the matrix A with the n number of rows and the m number of columns when it fulfils the conditions of:
i) no transversal T of the matrix A such that C z is contained in the set of row indices of T, and
ii) for each element c of C z there is a transversal T of the matrix A such that C z \{c} is wholly contained in the set of the row indices of T, and
the column rank comprises a set C s of natural numbers i, where 1≦i≦m, of the matrix A with the n number of rows and the m number of columns when it fulfils the conditions of:
i) no transversal T of the matrix A such that C s is contained in the set of row indices of T, and
ii) for each element c of C s there is a transversal T of the matrix A such that C s \{c} is wholly contained in the set of the column indices of T.
2. The method according to claim 1 , which further comprises:
prior to executing step 1:
applying an equation significance list of length of the n number in which each equation of the system of equations is assigned at least one of an equation number and an item of equation text information; and
applying a component significance list of length of the m number in which each component of a solution vector x is assigned at least one of a component number and an item of component text information;
in step 3, outputting at least one of the equation number and the item of equation text information in accordance with the equation significance list instead of outputting the numbers contained in each row rank; and
in step 3, outputting at least one of the component number and the item of component text information in accordance with the component significance list instead of outputting the numbers contained in each column rank.
3. The method of claim 1 , further comprising the step of downloading a computer program product or a computer program with computer-executable instructions for executing a method for numerical simulation of a technical system according to claim 1 from an electronic data network onto a computer connected to the data network.
4. The method according to claim 3 , wherein the electronic data network is the Internet.
5. A computer program product for simulating an electrical network comprising a computer memory, including computer executable instructions to perform a method for providing error information relating to inconsistencies, whereby the inconsistencies in a system of differential equations of the electrical network are detected with the aid of a computer or with the aid of an analog computer and the system of equations is of the form f (t, x (t), {dot over (x)} (t), . . . , x (k) (t), p )= 0 , including:
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where x (t) and derivatives {dot over (x)} (t), . . . , x (k) (t) thereof respectively have m number of_elements, and p is a parameter vector that can occur in the system of equations; and
the method comprising the following steps:
step 1: setting up a dependence matrix A with m number of columns and n number of rows, and setting an element A to
A (i, j)≠0 when an i th row of f (t, x (t), {dot over (x)} (t), . . . , x (k) (t),p) defined with f i (t, x (t), {dot over (x)} (t), . . . , x (k) is a function of
a) a j th element of x expressed as x j (t); or
b) one of the derivatives of the j th element of x defined as x j (s) (t);
and otherwise setting the element A (i, j)=0;
step 2: determining a set of row ranks each having numbers of those rows of the dependence matrix A that are mutually dependent and determining a set of column ranks of the dependence matrix A each having numbers of those columns of the dependence matrix A that are mutually dependent if such row ranks and/or column ranks are present;
step 3: outputting error information containing the numbers contained in each row rank determined in step 2 and in each column rank determined in step 2; and
the row rank includes a set C z of natural numbers i, where 1≦i≦n, of the matrix A with the n number of rows and the m number of columns when it fulfils the conditions of:
i) no transversal T of the matrix A such that C z is contained in the set of row indices of T, and
ii) for each element c of C z there is a transversal T of the matrix A such that C z \{c} is wholly contained in the set of the row indices of T, and
the column rank comprises a set C s of natural numbers i, where 1≦i≦m, of the matrix A with the n number of rows and the m number of columns when it fulfils the conditions of:
i) no transversal T of the matrix A such that C s is contained in the set of row indices of T, and
ii) for each element c of C s there is a transversal T of the matrix A such that C s \{c} is wholly contained in the set of the column indices of T.
6. The computer program product according to claim 5 , wherein the method for providing error information relating to inconsistencies further comprises:
prior to executing step 1:
applying an equation significance list of length of the n number in which each equation of the system of equations is assigned at least one of an equation number and an item of equation text information; and
applying a component significance list of length of the m number in which each component of a solution vector x is assigned at least one of a component number and an item of component text information;
in step 3, outputting at least one of the equation number and the item of equation text information in accordance with the equation significance list instead of outputting the numbers contained in each row rank; and
in step 3, outputting at least one of the component number and the item of component text information in accordance with the component significance list instead of outputting the numbers contained in each column rank.