/**
 * A complex data of the shape: `a + bi` which has a real component, and an imaginary
 * component. Both are represented with Javascript's 64-bit floating point numbers.
 *
 * @since 1.0.0
 */
import * as Eq_ from 'fp-ts/Eq';
import * as IO from 'fp-ts/IO';
import * as Mg from 'fp-ts/Magma';
import * as Mn from 'fp-ts/Monoid';
import * as Sg from 'fp-ts/Semigroup';
import * as Fld from 'fp-ts/Field';
import * as Sh from 'fp-ts/Show';
import * as O from 'fp-ts/Option';
import * as Auto from './Automorphism';
import * as Iso from './Iso';
import * as M from './Matrix';
import * as Poly from './Polynomial';
import * as TC from './typeclasses';
import * as V from './Vector';
import * as Inf from './infix';
/**
 * @since 1.0.0
 * @category Model
 */
export interface Complex {
    readonly Re: number;
    readonly Im: number;
}
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const zero: Complex;
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const one: Complex;
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const scalar: (a: number) => Complex;
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const of: (Re: number, Im: number) => Complex;
/**
 * Converts a polar-form complex tuple to `Complex`
 *
 * Note, psi here is in radians
 *
 * @since 1.0.0
 * @category Constructors
 */
export declare const fromPolarRadians: (r: number, psi: number) => Complex;
/**
 * Converts a polar-form complex tuple to `Complex`
 *
 * Note, theta here is in degrees
 *
 * @since 1.0.0
 * @category Constructors
 */
export declare const fromPolarDegrees: (r: number, theta: number) => Complex;
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const fromVector: (v: V.Vec<2, number>) => Complex;
/**
 * @since 1.0.0
 * @category Constructors
 */
export declare const randComplex: (min: number, max: number) => IO.IO<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const Eq: Eq_.Eq<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const MagmaSub: Mg.Magma<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const SemigroupSum: Sg.Semigroup<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const SemigroupProduct: Sg.Semigroup<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const MonoidSum: Mn.Monoid<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const MonoidProduct: Mn.Monoid<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const Field: Fld.Field<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const Show: Sh.Show<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const ComplexAdditiveAbelianGroup: TC.AbelianGroup<Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const ComplexBimodule: TC.Bimodule<Complex, number>;
/**
 * @since 1.0.0
 * @category Complex Ops
 */
export declare const conj: (c: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Complex Ops
 */
export declare const sqrt: (c: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Complex Ops
 */
export declare const pow: (c: Complex, n: number) => Complex;
/**
 * @since 1.0.0
 * @category Infix
 */
export declare const _: (a: Complex, s: Inf.FieldSymbol, b: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Infix
 */
export declare const $_: (s: Inf.FieldSymbol, x: Complex, y: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Infix
 */
export declare const _$: (a: Complex, b: Complex, s: Inf.FieldSymbol) => Complex;
/**
 * @since 1.0.0
 * @category Destructors
 */
export declare const toVector: (c: Complex) => V.Vec<2, number>;
/**
 * @since 1.0.0
 * @category Destructors
 */
export declare const argumentRadians: (c: Complex) => O.Option<number>;
/**
 * @since 1.0.0
 * @category Destructors
 */
export declare const argumentDegrees: (c: Complex) => O.Option<number>;
/**
 * @since 1.0.0
 * @category Destructors
 */
export declare const modulus: (c: Complex) => number;
/**
 * @since 1.0.0
 * @category Isomorphisms
 */
export declare const IsoVector: Iso.Iso<Complex, V.Vec<2, number>>;
/**
 * @since 1.0.0
 * @category Model
 */
export declare type Vec<N> = V.Vec<N, Complex>;
/**
 * @since 1.1.0
 * @category Constructors
 */
export declare const onesN: <N extends number>(n: N) => Vec<N>;
/**
 * @since 1.1.0
 * @category Constructors
 */
export declare const zerosN: <N extends number>(n: N) => Vec<N>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const AdditiveAbGrpN: <N extends number>(n: N) => TC.AbelianGroup<V.Vec<N, Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const BiModN: <N extends number>(n: N) => TC.Bimodule<V.Vec<N, Complex>, Complex, Complex>;
/**
 * Add two vectors
 *
 * @since 1.1.0
 * @category Matrix Operations
 */
export declare const addV: <N>(x: V.Vec<N, Complex>, y: V.Vec<N, Complex>) => V.Vec<N, Complex>;
/**
 * Subtract two vectors
 *
 * @since 1.1.0
 * @category Matrix Operations
 */
export declare const subV: <N>(x: V.Vec<N, Complex>, y: V.Vec<N, Complex>) => V.Vec<N, Complex>;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const dot: <N>(x: V.Vec<N, Complex>, y: V.Vec<N, Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const cross: (x: V.Vec<3, Complex>, y: V.Vec<3, Complex>) => V.Vec<3, Complex>;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const outerProduct: <M extends number, N extends number>(v1: V.Vec<M, Complex>, v2: V.Vec<N, Complex>) => M.Mat<M, N, Complex>;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const l1Norm: <N>(x: V.Vec<N, Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const l2Norm: <N>(x: V.Vec<N, Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const lpNorm: (p: number) => <N>(v: V.Vec<N, Complex>) => Complex;
/**
 * @since 1.1.0
 * @category Vector Operations
 */
export declare const lInfNorm: (v: V.Vec<unknown, Complex>) => number;
/**
 * @since 1.0.0
 * @category Vector Operations
 */
export declare const projection: <N extends number>(u: V.Vec<N, Complex>, v: V.Vec<N, Complex>) => V.Vec<N, Complex>;
/**
 * @since 1.0.0
 * @category Model
 */
export declare type Mat<M, N> = M.Mat<M, N, Complex>;
/**
 * @since 1.1.0
 * @category Instances
 */
export declare const MonoidProductMM: <M extends number>(m: M) => Mn.Monoid<M.Mat<M, M, Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const AdditiveAbGrpMN: <M extends number, N extends number>(m: M, n: N) => TC.AbelianGroup<M.Mat<M, N, Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const BiModMN: <M extends number, N extends number>(m: M, n: N) => TC.Bimodule<M.Mat<M, N, Complex>, Complex, Complex>;
/**
 * @since 1.0.0
 * @category Matrix Operations
 */
export declare const idMat: <M extends number>(m: M) => M.Mat<M, M, Complex>;
/**
 * Multiply two matricies with matching inner dimensions
 *
 * ```math
 * (A ∈ R_mn) (B ∈ R_np) = C ∈ R_mp
 * ```
 *
 * Efficiency: `(8mpn)` flops
 *
 * @since 1.0.0
 * @category Matrix Operations
 */
export declare const mulM: <M extends number, N1 extends number, N2 extends N1, P extends number>(x: M.Mat<M, N1, Complex>, y: M.Mat<N2, P, Complex>) => M.Mat<M, P, Complex>;
/**
 * Add two matricies
 *
 * @since 1.1.0
 * @category Matrix Operations
 */
export declare const addM: <M, N>(x: M.Mat<M, N, Complex>, y: M.Mat<M, N, Complex>) => M.Mat<M, N, Complex>;
/**
 * Subtract two matricies
 *
 * @since 1.1.0
 * @category Matrix Operations
 */
export declare const subM: <M, N>(x: M.Mat<M, N, Complex>, y: M.Mat<M, N, Complex>) => M.Mat<M, N, Complex>;
/**
 * Transform a column vector `x` into vector `b` by matrix `A`
 *
 * ```math
 * Ax = b
 * ```
 *
 * Efficiency: `8mn` flops
 *
 * @since 1.0.0
 * @category Matrix Operations
 */
export declare const linMap: <M, N1, N2 extends N1>(A: M.Mat<M, N1, Complex>, x: V.Vec<N2, Complex>) => V.Vec<M, Complex>;
/**
 * Transform a row-vector `x` into vector `b` by matrix `A`
 *
 * ```math
 * xA = b
 * ```
 *
 * Efficiency: `8mn` flops
 *
 * @since 1.1.0
 * @category Matrix Operations
 */
export declare const linMapR: <M extends number, N1 extends number, N2 extends N1>(x: V.Vec<N1, Complex>, A: M.Mat<N2, M, Complex>) => V.Vec<M, Complex>;
/**
 * @since 1.0.0
 * @category Matrix Operations
 */
export declare const trace: <M extends number>(fa: M.Mat<M, M, Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const PolynomialAdditiveAbelianGroup: TC.AbelianGroup<Poly.Polynomial<Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const PolynomialBimodule: TC.Bimodule<Poly.Polynomial<Complex>, Complex, Complex>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const PolynomialRing: TC.CommutativeRing<Poly.Polynomial<Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const PolynomialEuclidianRing: TC.EuclidianRing<Poly.Polynomial<Complex>>;
/**
 * @since 1.0.0
 * @category Instances
 */
export declare const getDifferentialAutomorphism: (constantTerm: Complex) => Auto.Automorphism<Poly.Polynomial<Complex>>;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const evaluatePolynomial: (p: Poly.Polynomial<Complex>) => (x: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const derivative: (coeffs: Poly.Polynomial<Complex>) => Poly.Polynomial<Complex>;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const integrate: (lower: Complex, upper: Complex) => (p: Poly.Polynomial<Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const getAntiderivative: (constantTerm: Complex) => (p: Poly.Polynomial<Complex>) => Poly.Polynomial<Complex>;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const polynomialInnerProduct: (p: Poly.Polynomial<Complex>, q: Poly.Polynomial<Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const polynomialNorm: (p: Poly.Polynomial<Complex>) => Complex;
/**
 * @since 1.0.0
 * @category Polynomial Operations
 */
export declare const polynomialProjection: (p: Poly.Polynomial<Complex>, q: Poly.Polynomial<Complex>) => Poly.Polynomial<Complex>;
/**
 * @since 1.0.0
 * @category Aliases
 */
export declare const add: (x: Complex, y: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Aliases
 */
export declare const sub: (x: Complex, y: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Aliases
 */
export declare const mul: (x: Complex, y: Complex) => Complex;
/**
 * @since 1.0.0
 * @category Aliases
 */
export declare const div: (x: Complex, y: Complex) => Complex;
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