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Chapter 5. Equations

algebraic loops and substitutes formulas with necessary equations. In our example it looks like:

0 = x + y0 = x − y −2 .

The solution is: x = 1, y = -1.

An algebraic loop also happens when initial values of two variables refer to each other. For example, when var1 initializes var2, var2 initializes var3, and var3 initializes var1. AnyLogic substitutes these expressions into an algebraic equation system, solves this system numerically, and initializes the variables with the obtained solution.

5.5 Runtime errors caused by equations

During time steps (in between discrete event steps), the model engine numerically solves the global set of equations. This process may cause errors of three types:

•A computational error in an expression specified by the user, e.g. division by zero or square root of a negative value.

•Incorrectly constructed set of equations, e.g. a variable appears more than once on the left-hand side.

•The set of equations cannot be solved with the given initial values.

In case of error of the third type, you should analyze the set of algebraic equations to find out if it has a solution. In case it has, it is recommended to give a good initial approximation to the solution. You should set up the initial approximation just before the set of equations is activated; e.g., in the entry action of the state containing equations.

5.6 Numerical methods

When a model starts, the equations are assembled into the main differential equation system. During the simulation, this DES is solved by one of the numerical methods built in AnyLogic. AnyLogic provides the rich collection of numerical methods for solving ordinal

Chapter 5. Equations

Time accuracy – the desired time accuracy for finding change events (switch points) when solving equations.

Fixed time step – fixed time step for methods using the fixed integration step (e.g. Euler, RK4).

If you do not specify particular solver, AnyLogic chooses the numerical solver automatically at runtime in accordance to the behavior of the system. For instance, when solving differential equations, it starts integration with forth-fifth-order Runge-Kutta method. If the system is stiff, AnyLogic plugs in another solver – RADAU5, designed for stiff differential equation systems. Both these methods change the integration step to achieve the given accuracy. You can set up methods with fixed step (Euler and forth-order Runge-Kutta), which are not as accurate as the first pair of methods, but perform numerical integration more rapidly. However, if you need a precise solution, use the Automatic solver.

Chapter 6. Matrices and hyper-arrays

►To set initial value for a matrix

1.Select the variable on the structure diagram.

2.In the Equations section of the Properties window, choose No equation from the Form drop-down list.

3.Specify the Initial Value expression.

Provide an initial value expression using such notation: delimit matrix with braces, enter value row-by-row, wrapping each row with braces and separating rows and elements in a row with commas. Be sure all the rows have the same number of elements. For example, a matrix is the following:

5 7.83 -1

0.2-3.4 0

For two-dimensional matrices these values are assigned in row-first order: Into the Initial value property, you type:

{{ 5, 7.83, -1}, { 0.2, -3.4, 0 }}

, or

[5, 7.83, -1 ; 0.2, -3.4, 0]

To increase readability you can split the string into multiple lines:

{{ 5, 7.83, -1 },

{ 0.2, -3.4, 0 }}

You can initialize identity matrices, i.e. matrices with ones on the main diagonal and zeros elsewhere, by typing identity keyword.

Chapter 6. Matrices and hyper-arrays

6.1.3.2 Modifying a matrix variable

Elements of a matrix can be modified during the model simulation.

►To modify an element of a matrix

1.Double-click the variable in the Model Explorer or on the animated structure diagram, or

Right-click the variable in the Model Explorer or on the animated structure diagram and choose Modify from the popup menu.

The Modify dialog box is displayed (see Figure 58).

2.In the Apply to section, specify whether you want to modify some specific item of a matrix, or all items once.

3.Type a new value in the New value edit box.

4.Click OK.

Figure 58. Modifying a matrix variable

6.1.4Accessing and modifying a matrix from code

6.1.4.1 Defining and initializing a matrix from code

To define a variable of matrix type in Java code (commonly in the Startup code section of the active object’s Code window), you use one of the following forms:

AnyLogic V User’s Manual

Matrix A = new Matrix( <number of rows>, <number of columns> ); to create a two-dimensional matrix the specified number of rows and columns

Matrix A = new Matrix( <integer number> );

to create a vector with the specified number of rows

Matrix A = new Matrix( <real number> );

to create a 1x1 matrix with the single element with the specified value

Matrix A = new Matrix ( <number of rows>, <number of columns>, <real number> )

to create a matrix initialized with a real number

Matrix A = Matrix.random( <number of rows>, <number of columns> ) to create a matrix with elements uniformly distributed in interval [0,1]

Matrix A = Matrix.identity( <number of rows>, <number of columns> ) to create an identity matrix with ones on the main diagonal and zeros elsewhere

Matrix A = new Matrix ( new double[][] { { 5, 7.83, -1 }, { 0.2, -3.4, 0 } } );

to create a 3x4 matrix initialized as shown in the previous section. Use the same notation as used for defining matrix variables in the Initial value property: delimit matrix with braces, enter value row-by-row, wrapping row with braces and separating rows and elements in a row with commas. Be sure all the rows hold the same number of elements.

To initialize matrix in code at once, write the following code:

If there is a simple algorithm evaluating the elements, you can initialize it using cycles. See the example below:

Write the following code to initialize this matrix:

int m = 3; // number of rows

Chapter 6. Matrices and hyper-arrays

int n = 4; // number of columns

A = new Matrix ( m, n ); // create zero matrix for (int i=0; i<m; i++){

for (int j=0; j<n; j++){

if (j >= i) A.set( i, j, 1-i+j ); else A.set( i, j, -5 );

}

}

6.1.4.2 Modifying and accessing elements of a matrix from code

If you want to operate with a matrix itself, you refer to it simply by its name.

If you need a submatrix (e.g., one of the columns) of the matrix A taking part in an equation, use the following function:

A.getMatrix(i0, i1, j0, j1)

where the points (i0,j0) and (i1,j1) specify the upper left and the lower right elements of the submatrix to select from the matrix A.

Please note that indexes in matrices are zero-based – i.e., start with “0”.

You can access the individual elements of matrices in the code by calling:

A.get(i,j) // if A is a matrix

To modify an element of a matrix, use the set(i,j,value) method, e.g.:

A = Matrix.identity(4,4);