Math.MFSolve

This module implements an equation solver that solves and evaluates expressions on the fly. It is based on Prof. D.E.Knuth's metafont. The goal of mfsolve is to make the solver useful in an interactive program, by enhancing the bidirectionality of the solver. Like metafont, it can solve linear equations, and evaluate nonlinear expressions. In addition to metafont, it also solves for angles, and makes the solution independend of the order of the equations.

The Expr datatype allows for calculations with constants and unknown variables. The Dependencies datatype contains all dependencies and known equations.

Examples:

Let's define some variables. The SimpleVar type is a simple wrapper around String to provide nice output, since the Show instance for String outputs quotation marks.

let [x, y, t, a] = map (makeVariable . SimpleVar) ["x", "y", "t", "a"]

Solve linear equations:

showVars $ flip execSolver noDeps $ do
  2*x + y === 5
  x - y   === 1

x = 2.0
y = 1.0

Solve for angle (pi/4):

showVars $ flip execSolver noDeps $ sin(t) === 1/sqrt(2)

t = 0.7853981633974484

Solve for angle (pi/3) and amplitude:

showVars $ flip execSolver noDeps $ do
  a*sin(x) === sqrt 3
  a*cos(x) === 1

x = 1.0471975511965979
a = 2.0

Allow nonlinear expression with unknown variables:

showVars $ flip execSolver noDeps $ do
  sin(sqrt(x)) === y
  x === 2

x = 2.0
y = 0.9877659459927355

Find the angle and amplitude when using a rotation matrix:

showVars $ flip execSolver noDeps $ do
  a*cos t*x - a*sin t*y === 30
  a*sin t*x + a*cos t*y === 40
  x === 10
  y === 10

x = 10.0
y = 10.0
t = 0.14189705460416402
a = 3.5355339059327373

Expressions

data SimpleExpr v n

A simplified datatype representing an expression. This can be used to inspect the structure of a Expr, which is hidden.

Constructors

SEBin BinaryOp (SimpleExpr v n) (SimpleExpr v n)

SEUn UnaryOp (SimpleExpr v n)

Var v

Const n

Instances

(Show v, Ord n, Show n, Num n, Eq n) => Show (SimpleExpr v n)

data Expr v n

A mathematical expression of several variables. Several Numeric instances (Num, Floating and Fractional) are provided, so doing calculations over Expr is more convenient.

Instances

(Floating n, Ord n, Ord v) => Floating (Expr v n)

(Floating n, Ord n, Ord v) => Fractional (Expr v n)

(Floating n, Ord n, Ord v) => Num (Expr v n)

(Ord n, Num n, Eq n, Show v, Show n) => Show (Expr v n)

Generic (Expr v n)

(Hashable v, Hashable n) => Hashable (Expr v n)

type Rep (Expr v n)

data LinExpr v n

A linear expression of several variables. For example: 2*a + 3*b + 2 would be represented as LinExpr 2 [(a, 2), (b, 3)].

Constructors

LinExpr n [(v, n)]

Instances

(Eq v, Eq n) => Eq (LinExpr v n)

(Show v, Show n) => Show (LinExpr v n)

Generic (LinExpr v n)

(Hashable v, Hashable n) => Hashable (LinExpr v n)

type Rep (LinExpr v n)

data UnaryOp

Constructors

Sin

Abs

Recip

Signum

Exp

Log

Cos

Cosh

Atanh

Tan

Sinh

Asin

Acos

Asinh

Acosh

Atan

Instances

Eq UnaryOp

Show UnaryOp

Generic UnaryOp

Hashable UnaryOp

type Rep UnaryOp

data BinaryOp

Constructors

Add

Mul

Instances

Eq BinaryOp

Show BinaryOp

newtype SimpleVar

Constructors

SimpleVar String

Instances

Eq SimpleVar

Ord SimpleVar

Show SimpleVar

A simple String wrapper, which will print formulas more cleanly.

Generic SimpleVar

Hashable SimpleVar

type Rep SimpleVar

makeVariable :: Num n => v -> Expr v n

Create an expression from a variable

makeConstant :: n -> Expr v n

Create an expression from a constant

evalExpr :: Floating n => (v -> n) -> SimpleExpr v n -> n

Evaluate the expression given a variable substitution.

fromSimple :: (Floating n, Ord n, Ord v) => SimpleExpr v n -> Expr v n

Make a expression from a simple expression.

toSimple :: (Num n, Eq n) => Expr v n -> SimpleExpr v n

Convert an Expr to a SimpleExpr.

evalSimple :: Floating m => (n -> m) -> (v -> m) -> SimpleExpr v n -> m

evaluate a simple expression using the given substitution.

hasVar :: (Num t, Eq v, Eq t) => v -> Expr v t -> Bool

The expression contains the given variable.

Dependencies

data Dependencies v n

This hidden datatype represents a system of equations. It contains linear dependencies on variables as well as nonlinear equations. The following terminology is used from metafont:

• known variable: A variable who's dependency is just a number.
• dependend variable: A variable which depends linearly on other variables.
• independend variable: any other variable.

A dependend variable can only depend on other independend variables. Nonlinear equations will be simplified by substituting and evaluating known variables, or by reducing some trigonometric equations to linear equations.

Instances

(Show n, Floating n, Ord n, Ord v, Show v) => Show (Dependencies v n)

Monad m => MonadState (Dependencies v n) (MFSolverT v n m)

data DepError v n

An error type for ===, =&= and addEquation:

Constructors

UndefinedVar v

The variable is not defined.

UnknownVar v n

The variable is defined but dependend an other variables.

InconsistentEq n

The equation was reduced to the impossible equation `a == 0` for nonzero a, which means the equation is inconsistent with previous equations.

RedundantEq

The equation was reduced to the redundant equation `0 == 0`, which means it doesn't add any information.

Instances

(Show n, Show v) => Show (DepError v n)

(Show v, Show n, Typeable * v, Typeable * n) => Exception (DepError v n)

Typeable (* -> * -> *) DepError

Monad m => MonadError (DepError v n) (MFSolverT v n m)

noDeps :: Dependencies v n

An empty system of equations.

addEquation :: (Hashable n, Hashable v, RealFrac (Phase n), Ord v, Floating n) => Dependencies v n -> Expr v n -> Either (DepError v n) (Dependencies v n)

addEquation d e: Add the equation e = 0 to the system d.

eliminate :: (Hashable n, Show n, Hashable v, RealFrac (Phase n), Ord v, Show v, Floating n) => Dependencies v n -> v -> (Dependencies v n, [Expr v n])

Eliminate an variable from the equations. Returns the eliminated equations. Before elimination it performs substitution to minimize the number of eliminated equations.

Important: this function is still experimental and mostly untested.

getKnown :: (Eq v, Hashable v) => v -> Dependencies v n -> Either [v] n

Return the value of the variable, or a list of variables it depends on. Only linear dependencies are shown.

knownVars :: Dependencies v n -> [(v, n)]

Return all known variables.

varDefined :: (Eq v, Hashable v) => v -> Dependencies v n -> Bool

Return True if the variable is known or dependend.

nonlinearEqs :: (Ord n, Ord v, Floating n) => Dependencies v n -> [Expr v n]

Return all nonlinear equations e_i, where e_i = 0.

dependendVars :: Eq n => Dependencies v n -> [(v, LinExpr v n)]

Return all dependend variables with their dependencies.

Monadic Interface

(===) :: (MonadState (Dependencies v n) m, MonadError (DepError v n) m, Eq v, Hashable v, Hashable n, RealFrac n, Floating n, Ord v) => Expr v n -> Expr v n -> m () infixr 1

Make the expressions on both sides equal

(=&=) :: (MonadState (Dependencies v n) m, MonadError (DepError v n) m, Eq v, Hashable v, Hashable n, RealFrac n, Floating n, Ord v) => (Expr v n, Expr v n) -> (Expr v n, Expr v n) -> m () infixr 1

Make the pairs of expressions on both sides equal. No error is signaled if the equation for one of the sides is Redundant for example in (x, 0) == (y, 0).

dependencies :: MonadState (Dependencies v n) m => m (Dependencies v n)

Get the dependencies from a state monad. Specialized version of get.

getValue :: (MonadState (Dependencies v n) m, MonadError (DepError v n) m, Eq v, Hashable v) => v -> m n

Return the value of the variable or throw an error.

getKnownM :: (MonadState (Dependencies v n) m, Hashable v, Eq v) => v -> m (Either [v] n)

Monadic version of getKnown.

varDefinedM :: (MonadState (Dependencies v n) m, Hashable v, Eq v) => v -> m Bool

Monadic version of varDefined.

eliminateM :: (MonadState (Dependencies v n) m, Hashable n, Hashable v, Show n, Show v, RealFrac n, Ord v, Floating n) => v -> m [Expr v n]

Monadic version of eliminate.

ignore :: MonadError (DepError v n) m => m () -> m ()

Succeed even when trowing a RedundantEq error.

MFSolver monad

type MFSolver v n a = MFSolverT v n Identity a

A monad for solving equations. Basicly just a state and exception monad over Dependencies and DepError.

runSolver :: MFSolver v n a -> Dependencies v n -> Either (DepError v n) (a, Dependencies v n)

run the solver.

evalSolver :: MFSolver v n a -> Dependencies v n -> Either (DepError v n) a

Return the result of solving the equations or an error.

execSolver :: MFSolver v n a -> Dependencies v n -> Either (DepError v n) (Dependencies v n)

Run the solver and return the dependencies or an error.

unsafeSolve :: (Typeable n, Typeable v, Show n, Show v) => Dependencies v n -> MFSolver v n a -> a

Return the result of solving the equations, or throw the error as an exception.

showVars :: (Show n, Show v, Ord n, Ord v, Floating n) => Either (DepError v n) (Dependencies v n) -> IO ()

Show all variables and equations. Useful in combination with execSolver.

MFSolverT monad transformer

data MFSolverT v n m a

A monad transformer for solving equations. Basicly just a state and exception monad transformer over Dependencies and DepError.

Instances

MonadReader s m => MonadReader s (MFSolverT v n m)

MonadWriter s m => MonadWriter s (MFSolverT v n m)

MonadTrans (MFSolverT v n)

Monad m => MonadError (DepError v n) (MFSolverT v n m)

Monad m => MonadState (Dependencies v n) (MFSolverT v n m)

Monad m => Monad (MFSolverT v n m)

Functor m => Functor (MFSolverT v n m)

(Monad m, Functor m) => Applicative (MFSolverT v n m)

MonadIO m => MonadIO (MFSolverT v n m)

MonadCont m => MonadCont (MFSolverT v n m)

runSolverT :: MFSolverT v n m a -> Dependencies v n -> m (Either (DepError v n) (a, Dependencies v n))

evalSolverT :: Functor f => MFSolverT v n f b -> Dependencies v n -> f (Either (DepError v n) b)

Return the result of solving the equations or an error. Monadic version.

execSolverT :: Functor m => MFSolverT v n m a -> Dependencies v n -> m (Either (DepError v n) (Dependencies v n))

Run the solver and return the dependencies or an error. Monadic version.

unsafeSolveT :: (Show n, Show v, Typeable n, Typeable v, Monad m) => Dependencies v n -> MFSolverT v n m a -> m a

Return the result of solving the equations, or throw the error as an exception. Monadic version.