All trigonometric functions are in radians (not degrees).

The available operands are listed below with a short description of each and a simple example involving variable x and y. Equations may be in the form of equality (=) or inequality (>,>=,<,<=) constraints. For inequalities, the equation may be bounded between lower and upper limits that are also functions of variables.

The available operands are listed below with a short description of each and a simple example involving variable x and y. Equations may be in the form of equalities (=) or inequalities (>,<). For inequalities, the equation may be bounded between lower and upper limits that are also functions of variables.

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     (y+2/x)^(x*z) * &
     (log(tanh(sqrt(y-x+x^2))+3))^2 &
     = 2+sinh(y)+acos(x+y)+asin(x/y)

The available operands are listed below with a short description of each and a simple example involving variable x and y. For equations may be in the form of equalities (=) or inequalities (>,<). For inequalities, the equation may be bounded between lower and upper limits that are also functions of variables.

There are currently 26 operands for parameters or variables. They are listed below with a short description of each and a simple example involving variable x and y. For equations may be in the form of equalities (=) or inequalities (>,<). For inequalities, the equation may be bounded between lower and upper limits that are also functions of variables.

A couple differential and algebraic equations are shown below. For steady-state solutions the differential variables ($x) are set to zero. Variables x, y, and z were not given initial values. In the absence of an initial condition, variables are set to a default value of 1.0.

 ! Example with three equality equations

  The steady-state solution is:
   p=2
   x=-1.0445
   y=0.1238
   z=-1.0445.  

(:cellnr:)

 ! Example with an inequality
 Model example
   Variables
     x
     y
     z
   End Variables

   Equations
     x = 0.5 * y
     0 = z + 2*x
     x < y < z
   End Equations
 End Model

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 ! Example model that demonstrates a few equations
 Model example
   Parameters
     p = 2
   End Parameters

   Variables
     x
     y
     z
   End Variables

   Equations
     exp(x*p)=y
     z = p*$x + x
     (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
   End Equations
 End Model

Operations

Example

A couple differential and algebraic equations are shown below. The steady-state solution is p=2, x=-1.0445, y=0.1238, and z=-1.0445. For steady-state solutions the differential variables ($x) are set to zero. Variables x, y, and z were not given initial values. In the absence of an initial condition, variables are set to a default value of 1.0.

A couple differential and algebraic equations are shown below. The steady-state solution is p=2, x=-1.0445, y=0.1238, and z=-1.0445. For steady-state solutions the differential variables ($x) are set to zero.

z = -1.0445

[@! Example model that demonstrates a few equations

Steady state solution

p = 2

x = -1.0445

y = 0.12380

z = -1.0445E+00

Open-equation format is allowed for differential and algebraic equations. Open-equation means that the equation can be expressed in the least restrictive form. Other software packages require differential equations to be posed in the semi-explicit form: dx/dt = f(x). This is not required with APMonitor modelling language. All equations are automatically transformed into residual form.

Equation Example

    p = 2

  Equations
    exp(x*p)=y
    z = p*$x + x
    (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
  End Equations

There are currently 21 operands for parameters or variables. They are listed below with a short description of each and a simple example involving variable x and y.

Equations consist of a collection of parameters and variables that are related by operands (+,-,*,/,exp(),d()/dt, etc.). The equations define the relationship between variables.

Open-equation format is allowed for differential and algebraic equations. Open-equation means that the equation can be expressed in the least restrictive form. Other software packages require differential equations to be posed in the semi-explicit form: dx/dt = f(x). This is not required with APMonitor modelling language.

Equation operands

There are currently 21 operands for parameters or variables. They are listed below with a short description of each and a simple example involving variable x and optionally y.

  Parameters
    p = 1
  End Parameters

  Variables
    x
    y
    z
  End Variables

Equations

Parameters are fixed values that represent model inputs, fixed constants, or any other value that does not change. Parameters are not modified by the solver as it searches for a solution. As such, parameters do not contribute to the number of degrees of freedom (DOF).

Equations are declared in the Equations ... End Equations section of the model file. The equations may be defined in one section or in multiple declarations throughout the model. Equations are parsed sequentially, from top to bottom. However, implicit equations are solved simultaneously so the order of the equations does not change the solution.

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! Example model that demonstrates equation declarations
Model example
 Variables
  x = 0.2
  y = 0.5
  z = 1.5
 End Variables

 Equations
  ! The program tranforms all equations from the 'original form' to
  !   the 'residual form'.  Sparse first derivatives
  !   of the residual are reported with respect to the variable values.
  x = y           ! Original form
  x-y = 0         ! Residual form

  ! Below are examples of some of the types of variable operations that
  !   are possible.  There is currently a limit of 100 unique variables per equation.
  -(x-y) = 0      ! Unary minus
  x+y=0           ! Addition   
  x-y=0           ! Subtraction   
  x*y=0           ! Multiplication   
  x/y=0           ! Division   
  x^y=0           ! Power   
  abs(x*y)=0      ! Absolute value   
  exp(x*y)=0      ! Exponentiation   
  log10(x*y)=0    ! Log10 
  log(x*y)=0      ! Log (natural log)   
  sqrt(x*y)=0     ! Square Root  
  sinh(x*y)=0     ! Hyperbolic Sine  
  cosh(x*y)=0     ! Hyperbolic Cosine  
  tanh(x*y)=0     ! Hyperbolic Tanget  
  sin(x*y)=0      ! Sine   
  cos(x*y)=0      ! Cosine   
  tan(x*y)=0      ! Tangent   
  asin(x*y)=0     ! Arc-sine  
  acos(x*y)=0     ! Arc-cos  
  atan(x*y)=0     ! Arc-tangent  

  ! Example of a more complex equation.  There are 3 unique variables (x,y,z) and 1 residual.
  ! Exact first derivatives are reported for:
  !   d(res)/dx, d(res)/dy, d(res)/dz
  ! where:
  !   res = (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 - (2+sinh(y)+acos(x+y)+asin(x/y))
  (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)

  ! Differential equation with $ indicating a differential with respect to time
  ! Sparsity pattern is augmented by n columns where n is the number of variables
  ! If x is the first variable and there are 3 variables then $x would be variable 4
  ! x=1
  ! y=2
  ! z=3
  ! $x=4
  ! $y=5
  ! $z=6
  $x = -x + y

  ! Characters are not case specific
  $Z = -x + z*Y
 End Equations
End Model

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