Main

Equations

Main.Equations History

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June 16, 2015, at 12:48 PM by 45.56.3.184 -
Deleted lines 0-1:
!! Equations
Changed line 48 from:
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
to:
(:table border=1 width=100% align=left bgcolor=#EEEEEE cellspacing=0:)
December 04, 2008, at 08:02 AM by 158.35.225.227 -
Deleted lines 21-23:
|| - ||Unary minus || -(x-y) = 0 ||
|| - ||Unary minus || -(x-y) = 0 ||
|| - ||Unary minus || -(x-y) = 0 ||
October 30, 2008, at 09:46 AM by 158.35.225.230 -
Added line 69:
Added line 71:
October 30, 2008, at 09:17 AM by 158.35.225.230 -
Changed lines 70-72 from:
     (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
to:
     (y+2/x)^(x*z) * &
   
(log(tanh(sqrt(y-x+x^2))+3))^2 &
   
= 2+sinh(y)+acos(x+y)+asin(x/y)
October 30, 2008, at 09:16 AM by 158.35.225.230 -
Changed line 11 from:
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.
to:
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.
October 30, 2008, at 09:14 AM by 158.35.225.230 -
Changed line 15 from:
|| !,#,% ||Comment ||!! equation #1 (:html:)<br>(:htmlend:) 0 = x[1] + x[2]  ! comment ||
to:
|| !,#,% ||Comment ||% equation #1 (:html:)<br>(:htmlend:) 0 = x[1] + x[2]  ! eqn1 ||
October 30, 2008, at 09:13 AM by 158.35.225.230 -
Changed line 15 from:
|| !,#,% ||Comment || ! equation #1 (:html:)<br>(:htmlend:) 0 = x[1] + x[2]  ! comment ||
to:
|| !,#,% ||Comment ||!! equation #1 (:html:)<br>(:htmlend:) 0 = x[1] + x[2]  ! comment ||
October 30, 2008, at 09:12 AM by 158.35.225.230 -
Changed line 15 from:
|| !,#,% ||Comment || ! comment ||
to:
|| !,#,% ||Comment || ! equation #1 (:html:)<br>(:htmlend:) 0 = x[1] + x[2] ! comment ||
October 30, 2008, at 09:11 AM by 158.35.225.230 -
Deleted line 15:
|| = ||Equality || x=y ||
October 30, 2008, at 09:10 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=50%
to:
|| border=1 width=80%
Changed lines 15-17 from:
|| = ||Line Continuation || 0 = x[1] & (:html:)<br>(:htmlend:) + x[2] ||
to:
|| !,#,% ||Comment || ! comment ||
|| = ||Equality || x=y ||
|| &
||Line Continuation || 0 = x[1] & (:html:)<br>(:htmlend:) + x[2] ||
October 30, 2008, at 09:09 AM by 158.35.225.230 -
Changed line 15 from:
|| = ||Line Continuation || 0 = x[1] & \n + x[2] ||
to:
|| = ||Line Continuation || 0 = x[1] & (:html:)<br>(:htmlend:) + x[2] ||
October 30, 2008, at 09:08 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=30%
to:
|| border=1 width=50%
Added line 15:
|| = ||Line Continuation || 0 = x[1] & \n + x[2] ||
October 28, 2008, at 03:16 PM by 158.35.225.230 -
Added lines 43-44:
|| erf() ||Error function || erf(x*y)=0 ||
|| erfc() ||Complementary error function || erfc(x*y)=0 ||
September 25, 2008, at 01:34 PM by 158.35.225.230 -
Changed lines 11-12 from:
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''.
to:
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.
Added lines 15-19:
|| = ||Equality || x=y ||
|| < ||Less than || x<y ||
|| <= ||Less than or equal || x<=y ||
|| > ||Greater than || x>y ||
|| >= ||Greater than or equal || x>=y ||
Added lines 21-23:
|| - ||Unary minus || -(x-y) = 0 ||
|| - ||Unary minus || -(x-y) = 0 ||
|| - ||Unary minus || -(x-y) = 0 ||
Changed lines 47-48 from:
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 (''$x'') are set to zero.  Variables x, y, and z were not given initial values.  In the absence of an initial condition, variables are set to a default value of 1.0.
to:
A couple differential and algebraic equations are shown below.  For steady-state solutions the differential variables (''$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.
Changed line 51 from:
 ! Example model that demonstrates a few equations
to:
 ! Example with three equality equations
Added lines 69-92:

  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

September 25, 2008, at 12:21 PM by 158.35.225.230 -
Changed lines 41-62 from:
(:table class='markup horiz' align='left':)
(:cellnr  class
='markup1':)
>>blue<<
[@!
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@]
>><<
to:
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
 !
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*$x + x
     (y+2/x)^(x*z) * (log(tanh(sqrt(y-x+x^2))+3))^2 = 2+sinh(y)+acos(x+y)+asin(x/y)
   End Equations
 End Model
September 25, 2008, at 10:08 AM by 158.35.225.230 -
Changed lines 9-10 from:
!!! Equation operands
to:
!!! Operations
Changed line 37 from:
!!! Equation Example
to:
!!! Example
September 25, 2008, at 09:52 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=30% align=left
to:
|| border=1 width=30%
September 25, 2008, at 09:52 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=30% align=center
to:
|| border=1 width=30% align=left
September 25, 2008, at 09:51 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=30%
to:
|| border=1 width=30% align=center
September 25, 2008, at 09:47 AM by 158.35.225.230 -
Changed line 14 from:
|| Operand ||! Description ||! Example ||
to:
||! Operand ||! Description ||! Example ||
September 25, 2008, at 09:42 AM by 158.35.225.230 -
Changed line 39 from:
A couple differential and algebraic equations are shown below.  The steady-state solution is p=2, x=-1.0445, y=0.1238, and z=-1.0445.  For steady-state solutions the differential variables (''$x'') are set to zero.
to:
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 (''$x'') are set to zero.  Variables x, y, and z were not given initial values.  In the absence of an initial condition, variables are set to a default value of 1.0.
September 25, 2008, at 09:40 AM by 158.35.225.230 -
Added lines 39-40:
A couple differential and algebraic equations are shown below.  The steady-state solution is p=2, x=-1.0445, y=0.1238, and z=-1.0445.  For steady-state solutions the differential variables (''$x'') are set to zero.
Deleted lines 44-48:
! Steady state solution
! p = 2
! x = -1.0445
! y = 0.12380
! z = -1.0445
September 25, 2008, at 09:37 AM by 158.35.225.230 -
Changed line 47 from:
! z = -1.0445E+00
to:
! z = -1.0445
September 25, 2008, at 09:37 AM by 158.35.225.230 -
Changed lines 42-47 from:
[@! Example model that demonstrates equation declarations
to:
[@! Example model that demonstrates a few equations
! Steady state solution
! p = 2
! x = -1.0445
! y = 0.12380
! z = -1.0445E+00
September 25, 2008, at 09:32 AM by 158.35.225.230 -
Changed lines 7-8 from:
''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 %blue%A%red%P%black%Monitor modelling language.
to:
''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 %blue%A%red%P%black%Monitor modelling language.  All equations are automatically transformed into residual form.
Changed lines 35-38 from:
|| $ ||Differential || $x = -x + y


to:
|| $ ||Differential || $x = -x + y ||

!!! Equation Example

Changed line 45 from:
   p = 1
to:
   p = 2
Changed lines 54-104 from:
 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
  $Z = -x + z*Y
 End Equations
to:
  Equations
    exp(x*p)=y
    z = p*$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
September 25, 2008, at 09:24 AM by 158.35.225.230 -
Added line 35:
|| $ ||Differential || $x = -x + y
September 25, 2008, at 09:24 AM by 158.35.225.230 -
Changed lines 14-34 from:
||! # ||! Description ||! Example ||
|| 1 ||Unary minus || -(x-y) = 0 ||
|| 2 ||Addition || x+y = 0 ||
|| 3 ||Subtraction || x-y=0 ||
|| 4 ||Multiplication || x*y=0 ||
|| 5 ||Division || x/y=0 ||
|| 6 ||Power || x^y=0 ||
|| 7 ||Absolute value || abs(x*y)=0 ||
|| 8 ||Exponentiation || exp(x*y)=0 ||
|| 9 ||Base-10 Log || log10(x*y)=0 ||
|| 10 ||Natural Log || log(x*y)=0 ||
|| 11 ||Square Root || sqrt(x*y)=0 ||
|| 12 ||Hyperbolic Sine || sinh(x*y)=0 ||
|| 13 ||Hyperbolic Cosine || cosh(x*y)=0 ||
|| 14 ||Hyperbolic Tanget || tanh(x*y)=0 ||
|| 15 ||Sine || sin(x*y)=0 ||
|| 16 ||Cosine || cos(x*y)=0 ||
|| 17 ||Tangent || tan(x*y)=0 ||
|| 18 ||Arc-sine || asin(x*y)=0 ||
|| 19 ||Arc-cos || acos(x*y)=0 ||
|| 20 ||Arc-tangent || atan(x*y)=0 ||
to:
|| Operand ||! Description ||! Example ||
|| - ||Unary minus || -(x-y) = 0 ||
|| + ||Addition || x+y = 0 ||
|| - ||Subtraction || x-y=0 ||
|| * ||Multiplication || x*y=0 ||
|| / ||Division || x/y=0 ||
|| ^ ||Power || x^y=0 ||
|| abs() ||Absolute value || abs(x*y)=0 ||
|| exp() ||Exponentiation || exp(x*y)=0 ||
|| log10 ||Base-10 Log || log10(x*y)=0 ||
|| log ||Natural Log || log(x*y)=0 ||
|| sqrt() ||Square Root || sqrt(x*y)=0 ||
|| sinh() ||Hyperbolic Sine || sinh(x*y)=0 ||
|| cosh() ||Hyperbolic Cosine || cosh(x*y)=0 ||
|| tanh() ||Hyperbolic Tanget || tanh(x*y)=0 ||
|| sin() ||Sine || sin(x*y)=0 ||
|| cos() ||Cosine || cos(x*y)=0 ||
|| tan() ||Tangent || tan(x*y)=0 ||
|| asin() ||Arc-sine || asin(x*y)=0 ||
|| acos() ||Arc-cos || acos(x*y)=0 ||
|| atan() ||Arc-tangent || atan(x*y)=0 ||
September 25, 2008, at 09:21 AM by 158.35.225.230 -
Changed line 14 from:
||! Description ||! Example ||
to:
||! # ||! Description ||! Example ||
September 25, 2008, at 09:20 AM by 158.35.225.230 -
Changed line 13 from:
|| border=1 width=50%
to:
|| border=1 width=30%
Changed lines 15-34 from:
|| Unary minus || -(x-y) = 0 ||
|| Addition || x+y = 0 ||
|| Subtraction || x-y=0 ||
|| Multiplication || x*y=0 ||
|| Division || x/y=0 ||
|| Power || x^y=0 ||
|| Absolute value || abs(x*y)=0 ||
|| Exponentiation
|| exp(x*y)=0 ||
|| Log10 || log10(x*y)=0 ||
|| Log (natural log) || log(x*y)=0 ||
|| Square Root || sqrt(x*y)=0 ||
|| Hyperbolic Sine
|| sinh(x*y)=0 ||
|| Hyperbolic Cosine || cosh(x*y)=0 ||
|| Hyperbolic Tanget || tanh(x*y)=0 ||
||
Sine || sin(x*y)=0 ||
|| Cosine || cos(x*y)=0 ||
|| Tangent || tan(x*y)=0 ||
|| Arc-sine || asin(x*y)=0 ||
|| Arc-cos || acos(x*y)=0 ||
|| Arc-tangent || atan(x*y)=0 ||
to:
|| 1 ||Unary minus || -(x-y) = 0 ||
|| 2 ||Addition || x+y = 0 ||
|| 3 ||Subtraction || x-y=0 ||
|| 4 ||Multiplication || x*y=0 ||
|| 5 ||Division || x/y=0 ||
|| 6 ||Power || x^y=0 ||
|| 7 ||Absolute value || abs(x*y)=0 ||
|| 8 ||Exponentiation || exp(x*y)=0 ||
|| 9 ||Base-10 Log || log10(x*y)=0 ||
|| 10 ||Natural Log || log(x*y)=0 ||
|| 11 ||Square Root || sqrt(x*y)=0 ||
|| 12 ||Hyperbolic Sine || sinh(x*y)=0 ||
|| 13 ||Hyperbolic Cosine || cosh(x*y)=0 ||
|| 14 ||Hyperbolic Tanget || tanh(x*y)=0 ||
|| 15 ||Sine || sin(x*y)=0 ||
|| 16 ||Cosine || cos(x*y)=0 ||
|| 17 ||Tangent || tan(x*y)=0 ||
|| 18 ||Arc-sine || asin(x*y)=0 ||
|| 19 ||Arc-cos || acos(x*y)=0 ||
|| 20 ||Arc-tangent || atan(x*y)=0 ||

September 25, 2008, at 09:17 AM by 158.35.225.230 -
Changed lines 11-12 from:
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''.
to:
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''.
Changed line 14 from:
||!Description || !Example ||
to:
||! Description ||! Example ||
September 25, 2008, at 09:16 AM by 158.35.225.230 -
Changed lines 14-34 from:
||!Description||!Example ||
||Unary minus|| -(x-y) = 0 ||
||Addition|| x+y = 0 ||
||Subtraction||x-y=0||
||Multiplication||x*y=0||
||Division||x/y=0||
||Power||x^y=0||
||Absolute value||abs(x*y)=0||
||Exponentiation||exp(x*y)=0||
||Log10||log10(x*y)=0||
||Log (natural log)||log(x*y)=0||
||Square Root||sqrt(x*y)=0||
||Hyperbolic Sine||sinh(x*y)=0||
||Hyperbolic Cosine||cosh(x*y)=0||
||Hyperbolic Tanget||tanh(x*y)=0||
||Sine||sin(x*y)=0||
||Cosine||cos(x*y)=0||
||Tangent||tan(x*y)=0||
||Arc-sine||asin(x*y)=0||
||Arc-cos||acos(x*y)=0||
||Arc-tangent||atan(x*y)=0||
to:
||!Description || !Example ||
|| Unary minus || -(x-y) = 0 ||
|| Addition || x+y = 0 ||
|| Subtraction || x-y=0 ||
|| Multiplication || x*y=0 ||
|| Division || x/y=0 ||
|| Power || x^y=0 ||
|| Absolute value || abs(x*y)=0 ||
|| Exponentiation || exp(x*y)=0 ||
|| Log10 || log10(x*y)=0 ||
|| Log (natural log) || log(x*y)=0 ||
|| Square Root || sqrt(x*y)=0 ||
|| Hyperbolic Sine || sinh(x*y)=0 ||
|| Hyperbolic Cosine || cosh(x*y)=0 ||
|| Hyperbolic Tanget || tanh(x*y)=0 ||
|| Sine || sin(x*y)=0 ||
|| Cosine || cos(x*y)=0 ||
|| Tangent || tan(x*y)=0 ||
|| Arc-sine || asin(x*y)=0 ||
|| Arc-cos || acos(x*y)=0 ||
|| Arc-tangent || atan(x*y)=0 ||
September 25, 2008, at 09:15 AM by 158.35.225.230 -
Changed lines 3-4 from:
Parameters are fixed values that represent model inputs, fixed constants, or any other value that does not changeParameters 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).
to:
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.
Added lines 7-36:
''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 %blue%A%red%P%black%Monitor 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''.

|| border=1 width=50%
||!Description||!Example ||
||Unary minus|| -(x-y) = 0 ||
||Addition|| x+y = 0 ||
||Subtraction||x-y=0||
||Multiplication||x*y=0||
||Division||x/y=0||
||Power||x^y=0||
||Absolute value||abs(x*y)=0||
||Exponentiation||exp(x*y)=0||
||Log10||log10(x*y)=0||
||Log (natural log)||log(x*y)=0||
||Square Root||sqrt(x*y)=0||
||Hyperbolic Sine||sinh(x*y)=0||
||Hyperbolic Cosine||cosh(x*y)=0||
||Hyperbolic Tanget||tanh(x*y)=0||
||Sine||sin(x*y)=0||
||Cosine||cos(x*y)=0||
||Tangent||tan(x*y)=0||
||Arc-sine||asin(x*y)=0||
||Arc-cos||acos(x*y)=0||
||Arc-tangent||atan(x*y)=0||

Changed lines 42-46 from:
 Variables
  x = 0.2
  y
= 0.5
  z = 1.5
End Variables
to:
  Parameters
 
   p = 1
  End Parameters

  Variables
    x
    y
    z
 
End Variables
September 25, 2008, at 08:58 AM by 158.35.225.230 -
Added lines 1-71:
!! 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.

(:table class='markup horiz' align='left':)
(:cellnr  class='markup1':)
>>blue<<
[@! 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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(:tableend:)