public static void TestSymbolism()
{
// Create some constants.
ComputerAlgebra.Expression A = 2;
ComputerAlgebra.Constant B = ComputerAlgebra.Constant.New(3);
// Create some variables.
ComputerAlgebra.Expression x = "x";
Variable y = Variable.New("y");
// Create basic expression with operator overloads.
ComputerAlgebra.Expression f = A * x + B * y + 4;
// This expression uses the implicit conversion from string to
// Expression, which parses the string.
ComputerAlgebra.Expression g = "5*x + C*y + 8";
// Create a system of equations from the above expressions.
var system = new List <Equal>()
{
Equal.New(f, 0),
Equal.New(g, 0),
};
// We can now solve the system of equations for x and y. Since the
// equations have a variable 'C', the solutions will not be
// constants.
List <Arrow> solutions = system.Solve(x, y);
Debug.WriteLine("The solutions are:");
foreach (Arrow i in solutions)
{
Debug.WriteLine(i.ToString());
}
}
static void Main(string[] args)
{
// Create some constants.
Expression A = 2;
Constant B = Constant.New(3);
// Create some variables.
Expression x = "x";
Variable y = Variable.New("y");
// Create a basic expressions.
Expression f = A * x + B * y + 4;
// This expression uses the implicit conversion from string to
// Expression, which parses the string.
Expression g = "5*x + C*y + 8";
// Create a system of equations from the above expressions.
var system = new List <Equal>()
{
Equal.New(f, 0),
Equal.New(g, 0),
};
// We can now solve the system of equations for x and y. Since the
// equations have a variable 'C', the solutions will not be
// constants.
List <Arrow> solutions = system.Solve(x, y);
Console.WriteLine("The solutions are:");
foreach (Arrow i in solutions)
{
Console.WriteLine(i.ToString());
}
// A fundamental building block of ComputerAlgebra is the 'Arrow'
// expression. Arrow expressions define the value of one expression
// to be the value given by another expression. For example, 'x->2'
// defines the expression 'x' to have the value '2'.
// The 'Solve' function used above returns a list of Arrow
// expressions defining the solutions of the system.
// Arrow expressions are used by the 'Evaluate' function to
// substitute values for expressions into other expressions. To
// demonstrate the usage of Evaluate, let's validate the solution
// by using Evaluate to substitute the solutions into the original
// system of equations, and then substitute a value for C.
Expression f_xy = f.Evaluate(solutions).Evaluate(Arrow.New("C", 2));
Expression g_xy = g.Evaluate(solutions).Evaluate(Arrow.New("C", 2));
if ((f_xy == 0) && (g_xy == 0))
{
Console.WriteLine("Success!");
}
else
{
Console.WriteLine("Failure! f = {0}, g = {1}", f_xy, g_xy);
}
// Suppose we need to evaluate the solutions efficiently many times.
// We can compile the solutions to delegates where 'C' is a
// parameter to the delegate, allowing it to be specified later.
var x_C = x.Evaluate(solutions).Compile <Func <double, double> >("C");
var y_C = y.Evaluate(solutions).Compile <Func <double, double> >("C");
for (int i = 0; i < 20; ++i)
{
double C = i / 2.0;
Console.WriteLine("C = {0}: (x, y) = ({1}, {2})", C, x_C(C), y_C(C));
}
}
/// <summary>
/// Solve the circuit for transient simulation.
/// </summary>
/// <param name="Analysis">Analysis from the circuit to solve.</param>
/// <param name="TimeStep">Discretization timestep.</param>
/// <param name="Log">Where to send output.</param>
/// <returns>TransientSolution describing the solution of the circuit.</returns>
public static TransientSolution Solve(Analysis Analysis, Expression TimeStep, IEnumerable <Arrow> InitialConditions, ILog Log)
{
Expression h = TimeStep;
Log.WriteLine(MessageType.Info, "Building solution for h={0}", TimeStep.ToString());
// Analyze the circuit to get the MNA system and unknowns.
List <Equal> mna = Analysis.Equations.ToList();
List <Expression> y = Analysis.Unknowns.ToList();
LogExpressions(Log, MessageType.Verbose, "System of " + mna.Count + " equations and " + y.Count + " unknowns = {{ " + String.Join(", ", y) + " }}", mna);
// Evaluate for simulation functions.
// Define T = step size.
Analysis.Add("T", h);
// Define d[t] = delta function.
Analysis.Add(ExprFunction.New("d", Call.If((0 <= t) & (t < h), 1, 0), t));
// Define u[t] = step function.
Analysis.Add(ExprFunction.New("u", Call.If(t >= 0, 1, 0), t));
mna = mna.Resolve(Analysis).OfType <Equal>().ToList();
// Find out what variables have differential relationships.
List <Expression> dy_dt = y.Where(i => mna.Any(j => j.DependsOn(D(i, t)))).Select(i => D(i, t)).ToList();
// Find steady state solution for initial conditions.
List <Arrow> initial = InitialConditions.ToList();
Log.WriteLine(MessageType.Info, "Performing steady state analysis...");
SystemOfEquations dc = new SystemOfEquations(mna
// Derivatives, t, and T are zero in the steady state.
.Evaluate(dy_dt.Select(i => Arrow.New(i, 0)).Append(Arrow.New(t, 0), Arrow.New(T, 0)))
// Use the initial conditions from analysis.
.Evaluate(Analysis.InitialConditions)
// Evaluate variables at t=0.
.OfType <Equal>(), y.Select(j => j.Evaluate(t, 0)));
// Solve partitions independently.
foreach (SystemOfEquations i in dc.Partition())
{
try
{
List <Arrow> part = i.Equations.Select(j => Equal.New(j, 0)).NSolve(i.Unknowns.Select(j => Arrow.New(j, 0)));
initial.AddRange(part);
LogExpressions(Log, MessageType.Verbose, "Initial conditions:", part);
}
catch (Exception)
{
Log.WriteLine(MessageType.Warning, "Failed to find partition initial conditions, simulation may be unstable.");
}
}
// Transient analysis of the system.
Log.WriteLine(MessageType.Info, "Performing transient analysis...");
SystemOfEquations system = new SystemOfEquations(mna, dy_dt.Concat(y));
// Solve the diff eq for dy/dt and integrate the results.
system.RowReduce(dy_dt);
system.BackSubstitute(dy_dt);
IEnumerable <Equal> integrated = system.Solve(dy_dt)
.NDIntegrate(t, h, IntegrationMethod.Trapezoid)
// NDIntegrate finds y[t + h] in terms of y[t], we need y[t] in terms of y[t - h].
.Evaluate(t, t - h).Cast <Arrow>()
.Select(i => Equal.New(i.Left, i.Right)).Buffer();
system.AddRange(integrated);
LogExpressions(Log, MessageType.Verbose, "Integrated solutions:", integrated);
LogExpressions(Log, MessageType.Verbose, "Discretized system:", system.Select(i => Equal.New(i, 0)));
// Solving the system...
List <SolutionSet> solutions = new List <SolutionSet>();
// Partition the system into independent systems of equations.
foreach (SystemOfEquations F in system.Partition())
{
// Find linear solutions for y. Linear systems should be completely solved here.
F.RowReduce();
IEnumerable <Arrow> linear = F.Solve();
if (linear.Any())
{
linear = Factor(linear);
solutions.Add(new LinearSolutions(linear));
LogExpressions(Log, MessageType.Verbose, "Linear solutions:", linear);
}
// If there are any variables left, there are some non-linear equations requiring numerical techniques to solve.
if (F.Unknowns.Any())
{
// The variables of this system are the newton iteration updates.
List <Expression> dy = F.Unknowns.Select(i => NewtonIteration.Delta(i)).ToList();
// Compute JxF*dy + F(y0) == 0.
SystemOfEquations nonlinear = new SystemOfEquations(
F.Select(i => i.Gradient(F.Unknowns).Select(j => new KeyValuePair <Expression, Expression>(NewtonIteration.Delta(j.Key), j.Value))
.Append(new KeyValuePair <Expression, Expression>(1, i))),
dy);
// ly is the subset of y that can be found linearly.
List <Expression> ly = dy.Where(j => !nonlinear.Any(i => i[j].DependsOn(NewtonIteration.DeltaOf(j)))).ToList();
// Find linear solutions for dy.
nonlinear.RowReduce(ly);
IEnumerable <Arrow> solved = nonlinear.Solve(ly);
solved = Factor(solved);
// Initial guess for y[t] = y[t - h].
IEnumerable <Arrow> guess = F.Unknowns.Select(i => Arrow.New(i, i.Evaluate(t, t - h))).ToList();
guess = Factor(guess);
// Newton system equations.
IEnumerable <LinearCombination> equations = nonlinear.Equations;
equations = Factor(equations);
solutions.Add(new NewtonIteration(solved, equations, nonlinear.Unknowns, guess));
LogExpressions(Log, MessageType.Verbose, String.Format("Non-linear Newton's method updates ({0}):", String.Join(", ", nonlinear.Unknowns)), equations.Select(i => Equal.New(i, 0)));
LogExpressions(Log, MessageType.Verbose, "Linear Newton's method updates:", solved);
}
}
Log.WriteLine(MessageType.Info, "System solved, {0} solution sets for {1} unknowns.",
solutions.Count,
solutions.Sum(i => i.Unknowns.Count()));
return(new TransientSolution(
h,
solutions,
initial));
}
public void AddEquation(Expression a, Expression b)
{
AddEquations(Equal.New(a, b));
}
