32. Applied Mathematics in J

Complex Numbers

All the mathematical functions can be applied to complex numbers. A complex constant is written with the letter j separating the real and imaginary parts, e. g. 0j1 is the square-root of _1 . Alternatively, a constant can be written in polar form in which the letters ar (or ad) separate the magnitude of the number from the angle in radians (or degrees) between the real axis and a line in the complex plane from the origin to the point representing the number:

1ar1

0.540302j0.841471

1ad90

0j1

A number of verbs are available for operations on components of complex numbers. All have rank 0.

+ y conjugate of y

+. y creates a 2-atom list of the real and imaginary components of y

. y creates a 2-atom list of the length and angle in radians of the polar form of y | y magnitude of y

xr. y x * r. y (i. e. polar form, where x is the magnitude and y the angle in radians)

! y factorial of y (more generally, the gamma function Γ(1+y))

Matrix Operations

J has primitive verbs for operations on matrices, some using the conjunction . . -/ .* y gives the determinant of y; x +/ .* y is the matrix product of x and y;

%. y is the matrix inverse of y (provided y is nonsingular); x %. y is the projection of x onto y .

In x +/ .* y, a rank-1 x is treated as a matrix with one row, and a rank-1 y is treated as a matrix with one column; but the result rank, which is 2 when rank-2 matrices are multiplied, is 1 when one operand has rank 1 and 0 when both do.

x %. y is a rough-and-ready way to get a least-squares approximation of x as a linear combination of the columns of y . If your y is singular or close to it, avoid %. and use methods based on singular value decomposition.

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Polynomials: p.

J supports 3 different ways of specifying a polynomial:

1.as a list of coefficients of increasing powers, starting with power 0 (i. e. a constant term), where each coefficient multiplied by the corresponding power of the variable produces a term; the polynomial is the sum of the terms. This form is an unboxed numeric list.

2.as a multiplier m and a list of roots r, representing the polynomial

m(x-r0)(x-r1)…(x-rn). This form is a 2-item list of boxes m;r (if m is 1 it may be omitted, leaving just <r).

3.(for multinomials) a multinomial of n variables is represented as a list of terms, each term consisting of a coefficient and a list of n exponents (one exponent per variable). The multinomial is the sum of the terms. The form is a boxed rank-2 array in which each item is a coefficient followed by the list of n exponents. This form is distinguished from the multiplier-root form by the rank of the boxed array. (It is possible to have multiple multinomials that share a common exponent array, by having more than one coefficient preceding each list of exponents, but we will not pursue that here)

For example, three ways of expressing the polynomial 3x3-12x (which can be written 3x(x+2)(x-2))are 0 _12 0 3, 3;_2 0 2, and <2 2$_12 1 3 3 .

p. y has rank 1 and converts between coefficient and multiplier-root form of the polynomial y . Note that converting from coefficient form to multiplier-root form solves for the roots of the polynomial.

p. 0 _12 0 3 +-+------+ |3|2 _2 0| +-+------+

p. 3;2 0 _2 0 _12 0 3

If the multinomial form has only one variable (i. e. each item has length 2), monad p. will convert it to coefficient form:

p. <2 2$_12 1 3 3 0 _12 0 3

Dyad p. has rank 1 0 and is used to evaluate a polynomial in any of these forms. If x is a polynomial in coefficient or multiplier-root form, x p. y evaluates it with y giving the value of the variable:

0 _12 0 3 p. 1

_9

(3;_2 0 2) p. _2 _1 0 1 2 0 9 0 _9 0

(the second evaluation applied the polynomial to 5 different values of the variable).

If x is a multinomial, x p. <y evaluates it with the list y giving the values of the variables (y must be boxed because the right rank of dyad p. is 0 and in this case there is

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more than one variable). So to evaluate the binomial x3+3x2y+3xy2+y3 with x=2 and y=3 we have

(<4 3$1 3 0 3 2 1 3 1 2 1 0 3) p. <2 3

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as expected.

Calculus: d., D., D:, and p..

J provides support for differential and integral calculus. u d. n produces the verb that gives the nth derivative of u :

*: d. 1

+:

*: y is y squared; the derivative is +: y which is y doubled.

^&3 d. 1 3&*@(^&2)

^&3 y is y cubed; the derivative is 3 * y ^ 2 .

^&3 d. 2 3"0 * +:

Second derivative is 3 * 2 * y .

*: d. _1

0 0 0 0.33333&p.

The _1st derivative is the indefinite integral, which is (y ^ 3) % 3 . The form the interpreter uses is the polynomial form.

f =. *:

f d. _1 (6)

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g =. *:@+: g d. _1 (6)

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u d. n produces ordinary derivatives, and evaluates its u with rank 0. For partial derivatives, use u D. n where u has rank greater than 0. Each cell of y produces an array of results, one result for each atom of the cell, giving the partial derivative with respect to that atom. For example, the length of a vector is given by

veclength =: +/&.:*:"1

which squares the atoms, adds them, and takes the square root: veclength 3 4 5

7.07107

The derivative of the vector length with respect to the individual components is given by: veclength D. 1 (3 4 5)

0.424264 0.565685 0.707107

The result of u need not be a scalar. Here we define the cross product:

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xp =: dyad : '((1|.x.)*(_1|.y.)) - ((_1|.x.)*(1|.y.))'"1 0 0 2&xp D. 1 (4 2 0)

0 2 0 _2 0 0 0 0 0

Each row is the vector-valued partial derivative of the cross product (0 0 2 xp 4 2 0) with respect to one component of y .

The interpreter will bend every effort to find a derivative for your function, but it may fail, or you may not like the derivative it chooses. m D. n, when m is a gerund u`v, produces a new verb which executes like u but whose nth derivative is v :

a =. *:`] D. 1

Here a is defined to be *: except that its derivative is ] . a

*:`]D.1

a D. 1 (5)

5

Sure enough, the derivative is ] .

If you don't want to express the derivative, you can have the interpreter approximate it for you. x u D: n y approximates the derivative at y by evaluating u at y and y+x . Both the left and right ranks of u D: n are the monadic rank of u, so you can specify different step-sizes for different atoms of y (if x is a scalar, it is used for the stepsize at all atoms of y).

You can do calculus on polynomials by manipulating the polynomial forms without having to create the verbs that operate on those forms. p.. y (rank 1) takes polynomial y in either coefficient or multiplier-root form and produces the coefficient form of the derivative of y . x p.. y (rank 0 1) produces the coefficient form of the integral of y with constant term x .

Taylor Series: t., t:, and T.

With derivatives available, Taylor series are a natural next step. u t. y is the yth Taylor coefficient of u expanded about 0, and x u t. y evaluates that term at the point x, in other words x u t. y is x^y * u t. y . All ranks of u t. are 0.

u T. n is a verb which is the n-term Taylor approximation to u (expanded about 0).

u t: y is (!y) * u t. y .

Hypergeometric Function with H.

The generalized hypergeometric function is specified by two lists of numbers, a numerator list and a denominator list. The generalized hypergeometric function is the sum over all k of the (infinite) generalized hypergeometric series, which is a power series in which the coefficient of the yk term is the product of the rising factorials of length k of the numerator items divided by a similar product for the denominator items, and then divided by !k .

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