Eq package:numeric-prelude

We need a Haskell 98 type class which provides equality test for Vector type constructors.

The Eq class defines equality (==) and inequality (/=). All the basic datatypes exported by the Prelude are instances of Eq, and Eq may be derived for any datatype whose constituents are also instances of Eq. The Haskell Report defines no laws for Eq. However, == is customarily expected to implement an equivalence relationship where two values comparing equal are indistinguishable by "public" functions, with a "public" function being one not allowing to see implementation details. For example, for a type representing non-normalised natural numbers modulo 100, a "public" function doesn't make the difference between 1 and 201. It is expected to have the following properties:

• Reflexivity x == x = True
• Symmetry x == y = y == x
• Transitivity if x == y && y == z = True, then x == z = True
• Substitutivity if x == y = True and f is a "public" function whose return type is an instance of Eq, then f x == f y = True
• Negation x /= y = not (x == y)

Minimal complete definition: either == or /=.

Two polynomials may be stored differently. This function checks whether two values of type LaurentPolynomial actually represent the same polynomial.

Lazy evaluation allows for the solution of differential equations in terms of power series. Whenever you can express the highest derivative of the solution as explicit expression of the lower derivatives where each coefficient of the solution series depends only on lower coefficients, the recursive algorithm will work.

Example for a linear equation: Setup a differential equation for y with

y   t = (exp (-t)) * (sin t)
y'  t = -(exp (-t)) * (sin t) + (exp (-t)) * (cos t)
y'' t = -2 * (exp (-t)) * (cos t)

Thus the differential equation

y'' = -2 * (y' + y)

holds. The following function generates a power series for exp (-t) * sin t by solving the differential equation.

We are not restricted to linear equations! Let the solution be y with y t = (1-t)^-1 y' t = (1-t)^-2 y'' t = 2*(1-t)^-3 then it holds y'' = 2 * y' * y

Some common quantity classes.

The value of seq a b is bottom if a is bottom, and otherwise equal to b. In other words, it evaluates the first argument a to weak head normal form (WHNF). seq is usually introduced to improve performance by avoiding unneeded laziness. A note on evaluation order: the expression seq a b does not guarantee that a will be evaluated before b. The only guarantee given by seq is that the both a and b will be evaluated before seq returns a value. In particular, this means that b may be evaluated before a. If you need to guarantee a specific order of evaluation, you must use the function pseq from the "parallel" package.

Evaluate each monadic action in the structure from left to right, and collect the results. For a version that ignores the results see sequence_.

Evaluate each monadic action in the structure from left to right, and ignore the results. For a version that doesn't ignore the results see sequence. As of base 4.8.0.0, sequence_ is just sequenceA_, specialized to Monad.