abs(x)

Returns the absolute value or vector length of x.

X = (x,y,z)
abs(X)

┌            ┐1/2
│ 2    2    2│   
│x  + y  + z │   
│            │   
└            ┘   

adj(m)

Returns the adjunct of matrix m. Adjunct is equal to determinant times inverse.

A = ((a,b),(c,d))
adj(A) == det(A) inv(A)

1

and(a,b,...)

Returns 1 if all arguments are true (nonzero). Returns 0 otherwise.

and(1=1,2=2)

1

arccos(x)

Returns the arc cosine of x.

arccos(1/2)

 1   
╶─╴ π
 3   

arccosh(x)

Returns the arc hyperbolic cosine of x.

arcsin(x)

Returns the arc sine of x.

arcsin(1/2)

 1   
╶─╴ π
 6   

arcsinh(x)

Returns the arc hyperbolic sine of x.

arctan(y,x)

Returns the arc tangent of y over x. If x is omitted then x = 1 is used.

arctan(1,0)

 1   
╶─╴ π
 2   

arctanh(x)

Returns the arc hyperbolic tangent of x.

arg(z)

Returns the angle of complex z.

arg(2 - 3i)

-arctan(3,2)

binding(s)

The result of evaluating a symbol can differ from the symbol's binding. For example, the result may be expanded. The binding function returns the actual binding of a symbol.

p = quote((x + 1)^2)
p

     2
p = x  + 2 x + 1

binding(p)

       2
(x + 1)

ceiling(x)

Returns the smallest integer greater than or equal to x.

ceiling(1/2)

1

check(x)

If x is true (nonzero) then continue in a script, else stop. Expression x can include the relational operators =, ==, <, <=, >, >=. Use the not function to test for inequality.

A = exp(i pi)
B = -1
check(A == B) -- stop here if A not equal to B

choose(n,k)

Returns the binomial coefficient n choose k.

choose(52,5) -- number of poker hands

2598960

circexp(x)

Returns expression x with circular and hyperbolic functions converted to exponentials.

circexp(cos(x) + i sin(x))

exp(i x)

clear

Clears all symbol definitions.

clock(z)

Returns complex z in polar form with base of negative 1 instead of e.

clock(2 - 3i)

            arctan(3,2) 
          -╶───────────╴
  1/2            π      
13    (−1)              

cofactor(m,i,j)

Returns the cofactor of matrix m for row i and column j.

A = ((a,b),(c,d))
cofactor(A,1,2) == adj(A)[2,1]

1

conj(z)

Returns the complex conjugate of z.

conj(2 - 3i)

2 + 3 i

contract(a,i,j)

Returns tensor a summed over indices i and j. If i and j are omitted then 1 and 2 are used. The expression contract(m) computes the trace of matrix m.

A = ((a,b),(c,d))
contract(A)

a + d

cos(x)

Returns the cosine of x.

cos(pi/4)

  1   
╶────╴
  1/2 
 2    

cosh(x)

Returns the hyperbolic cosine of x.

circexp(cosh(x))

 1             1        
╶─╴ exp(−x) + ╶─╴ exp(x)
 2             2        

cross(u,v)

Returns the cross product of vectors u and v.

curl(v)

Returns the curl of vector v with respect to symbols x, y, and z.

d(f,x,...)

Returns the derivative of f with respect to x and any additional arguments.

d(sin(x),x)

cos(x)

d(sin(x),x,x)

−sin(x)

d(sin(x y),x,2,y,2)

 2  2                                       
x  y  sin(x y) − 4 x y cos(x y) − 2 sin(x y)

Argument f can be a tensor of any rank. Argument x can be a vector. When x is a vector the result is the gradient of f.

F = (f(),g(),h())
X = (x,y,z)
d(F,X)

┌                                ┐
│ d(f(),x)   d(f(),y)   d(f(),z) │
│                                │
│ d(g(),x)   d(g(),y)   d(g(),z) │
│                                │
│ d(h(),x)   d(h(),y)   d(h(),z) │
└                                ┘

Symbol d can be used as a variable name. Doing so does not conflict with function d.

Symbol d can be redefined as a different function. The function derivative, a synonym for d, can be used to obtain a partial derivative.

defint(f,x,a,b)

Returns the definite integral of f with respect to x evaluated from a to b. The argument list can be extended for multiple integrals as shown in the following example.

f = (1 + cos(theta)^2) sin(theta)
defint(f, theta, 0, pi, phi, 0, 2 pi) -- integrate over theta then over phi

 16   
╶──╴ π
 3    

denominator(x)

Returns the denominator of expression x.

denominator(a/b)

b

det(m)

Returns the determinant of matrix m.

A = ((a,b),(c,d))
det(A)

a d - b c

dim(a,n)

Returns the dimension of the nth index of tensor a. Index numbering starts with 1.

A = ((1,2),(3,4),(5,6))
dim(A,1)

3

div(v)

Returns the divergence of vector v with respect to symbols x, y, and z.

do(a,b,...)

Evaluates each argument from left to right. Returns the result of the final argument.

do(A=1,B=2,A+B)

3

dot(a,b,...)

Returns the dot product of vectors, matrices, and tensors. Also known as the matrix product. Arguments are evaluated from right to left. The following example solves for X in AX = B.

A = ((1,2),(3,4))
B = (5,6)
X = dot(inv(A),B)
X

    ┌     ┐
    │ −4  │
    │     │
X = │  9  │
    │ ╶─╴ │
    │  2  │
    └     ┘

draw(f,x)

Draws a graph of f(x). Drawing ranges can be set with xrange and yrange.

xrange = (0,1)
yrange = (0,1)
draw(x^2,x)

eigenvec(m)

Returns eigenvectors for matrix m. Matrix m is required to be numerical, real, and symmetric. The return value is a matrix with each column an eigenvector. Eigenvalues are obtained as shown.

A = ((1,2),(3,4))
A = A + transpose(A) -- symmetrize
Q = eigenvec(A)
D = dot(transpose(Q),A,Q) -- eigenvalues are on the diagonal of D
dot(Q,D,transpose(Q))

    ┌       ┐
    │ 2   5 │
A = │       │
    │ 5   8 │
    └       ┘

eval(f,x,a,y,b,...)

Returns f evaluated with x replaced by a, y replaced by b, etc. All arguments can be expressions.

f = sqrt(x^2 + y^2)
eval(f,x,3,y,4)

5

In the following example, eval is used to replace x with cos(theta).

-- associated legendre of cos theta
P(l,m,x) = test(m < 0, (-1)^m (l + m)! / (l - m)! P(l,-m),
           1 / (2^l l!) sin(theta)^m *
           eval(d((x^2 - 1)^l, x, l + m), x, cos(theta)))

P(2,-1)

  1
−╶─╴ cos(θ) sin(θ)
  2

exp(x)

Returns the exponential of x.

exp(i pi)

-1

expcos(z)

Returns the cosine of z in exponential form.

expcos(z)

 1              1           
╶─╴ exp(i z) + ╶─╴ exp(−i z)
 2              2           

expcosh(z)

Returns the hyperbolic cosine of z in exponential form.

expcosh(z)

 1             1        
╶─╴ exp(−z) + ╶─╴ exp(z)
 2             2        

expsin(z)

Returns the sine of z in exponential form.

expsin(z)

  1                1             
−╶─╴ i exp(i z) + ╶─╴ i exp(−i z)
  2                2             

expsinh(z)

Returns the hyperbolic sine of z in exponential form.

expsinh(z)

  1             1        
−╶─╴ exp(−z) + ╶─╴ exp(z)
  2             2        

exptan(z)

Returns the tangent of z in exponential form.

exptan(z)

       i             i exp(2 i z)  
╶──────────────╴ − ╶──────────────╴
 exp(2 i z) + 1     exp(2 i z) + 1 

exptanh(z)

Returns the hyperbolic tangent of z in exponential form.

exptanh(z)

       1             exp(2 z)   
−╶────────────╴ + ╶────────────╴
  exp(2 z) + 1     exp(2 z) + 1 

factorial(n)

Returns the factorial of n. The expression n! can also be used.

20!

2432902008176640000

float(x)

Returns expression x with rational numbers and integers converted to floating point values. The symbol pi and the natural number are also converted.

float(212^17)

          39
3.52947 10

floor(x)

Returns the largest integer less than or equal to x.

floor(1/2)

0

for(i,j,k,a,b,...)

For i equals j through k evaluate a, b, etc.

for(k,1,3,A=k,print(A))

A = 1
A = 2
A = 3

Note: The original value of i is restored after for completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of for.

grad(f)

Returns the gradient d(f,(x,y,z)).

grad(f())

┌          ┐
│ d(f(),x) │
│          │
│ d(f(),y) │
│          │
│ d(f(),z) │
└          ┘

hadamard(a,b,...)

Returns the Hadamard (element-wise) product. The arguments are required to have the same dimensions. The Hadamard product is also accomplished by simply multiplying the arguments.

A = ((A11,A12),(A21,A22))
B = ((B11,B12),(B21,B22))
A B

┌                   ┐
│ A   B     A   B   │
│  11  11    12  12 │
│                   │
│ A   B     A   B   │
│  21  21    22  22 │
└                   ┘

i

Symbol i is initialized to the imaginary unit (−1)^1/2.

exp(i pi)

-1

Note: It is OK to clear or redefine i and use the symbol for something else.

imag(z)

Returns the imaginary part of complex z.

imag(2 - 3i)

-3

infixform(x)

Converts expression x to a string and returns the result.

p = (x + 1)^2
infixform(p)

x^2 + 2 x + 1

inner(a,b,...)

Returns the inner product of vectors, matrices, and tensors. Also known as the matrix product. Arguments are evaluated from right to left.

A = ((a,b),(c,d))
B = (x,y)
inner(A,B)

┌           ┐
│ a x + b y │
│           │
│ c x + d y │
└           ┘

Note: inner and dot are the same function.

integral(f,x)

Returns the integral of f with respect to x.

integral(x^2,x)

 1   3
╶─╴ x 
 3    

inv(m)

Returns the inverse of matrix m.

A = ((1,2),(3,4))
inv(A)

┌            ┐
│ −2     1   │
│            │
│  3      1  │
│ ╶─╴   −╶─╴ │
│  2      2  │
└            ┘

j

Set j=sqrt(-1) to use j for the imaginary unit instead of i.

j = sqrt(-1)
1/sqrt(-1)

-j

kronecker(a,b,...)

Returns the Kronecker product of vectors and matrices.

A = ((1,2),(3,4))
B = ((a,b),(c,d))
kronecker(A,B)

┌                       ┐
│  a     b    2 a   2 b │
│                       │
│  c     d    2 c   2 d │
│                       │
│ 3 a   3 b   4 a   4 b │
│                       │
│ 3 c   3 d   4 c   4 d │
└                       ┘

last

The result of the previous calculation is stored in last.

212^17

3529471145760275132301897342055866171392

last^(1/17)

212

Note: Symbol last is an implied argument when a function has no argument list.

212^17

3529471145760275132301897342055866171392

float

          39
3.52947 10

log(x)

Returns the natural logarithm of x.

log(x^y)

y log(x)

mag(z)

Returns the magnitude of complex z.

mag(x + i y)

┌       ┐1/2
│ 2    2│   
│x  + y │   
│       │   
└       ┘   

minor(m,i,j)

Returns the minor of matrix m for row i and column j.

A = ((1,2,3),(4,5,6),(7,8,9))
minor(A,1,1) == det(minormatrix(A,1,1))

1

minormatrix(m,i,j)

Returns a copy of matrix m with row i and column j removed.

A = ((1,2,3),(4,5,6),(7,8,9))
minormatrix(A,1,1)

┌       ┐
│ 5   6 │
│       │
│ 8   9 │
└       ┘

mod(a,b)

Returns the remainder of a over b.

mod(5,3/8)

 1 
╶─╴
 8 

noexpand(x)

Evaluates expression x without expanding products of sums.

noexpand((x + 1)^2 / (x + 1))

x + 1

not(x)

Returns 0 if x is true (nonzero). Returns 1 otherwise.

not(1=1)

0

nroots(p,x)

Returns the approximate roots of polynomials with real or complex coefficients. Multiple roots are returned as a vector. See also roots.

p = x^5 - 1
nroots(p,x)

┌                        ┐
│           1            │
│                        │
│ −0.809017 + 0.587785 i │
│                        │
│ −0.809017 − 0.587785 i │
│                        │
│ 0.309017 + 0.951057 i  │
│                        │
│ 0.309017 − 0.951057 i  │
└                        ┘

numerator(x)

Returns the numerator of expression x.

numerator(a/b)

a

or(a,b,...)

Returns 1 if at least one argument is true (nonzero). Returns 0 otherwise.

or(1=1,2=2)

1

outer(a,b,...)

Returns the outer product of vectors, matrices, and tensors.

A = (a,b,c)
B = (x,y,z)
outer(A,B)

┌                 ┐
│ a x   a y   a z │
│                 │
│ b x   b y   b z │
│                 │
│ c x   c y   c z │
└                 ┘

pi

Symbol for π.

exp(i pi)

-1

polar(z)

Returns complex z in polar form.

polar(x - i y)

┌       ┐1/2                    
│ 2    2│                       
│x  + y │    exp(-i arctan(y,x))
│       │                       
└       ┘                       

power

Use ^ to raise something to a power. Use parentheses for negative powers.

x^(-1/2)

  1   
╶────╴
  1/2 
 x    

print(a,b,...)

Evaluate expressions and print the results. Useful for printing from inside a for loop.

for(j,1,3,print(j))

j = 1
j = 2
j = 3

product(i,j,k,f)

For i equals j through k evaluate f. Returns the product of all f.

product(j,1,3,x + j)

 3      2
x  + 6 x  + 11 x + 6

The original value of i is restored after product completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of product.

product(y)

Returns the product of components of y.

y = (1,2,3,4)
product(y)

24

quote(x)

Returns expression x without evaluating it first.

quote((x + 1)^2)

       2
(x + 1)

rank(a)

Returns the number of indices that tensor a has.

A = ((a,b),(c,d))
rank(A)

2

rationalize(x)

Returns expression x with everything over a common denominator.

rationalize(1/a + 1/b + 1/2)

 2 a + a b + 2 b 
╶───────────────╴
      2 a b      

Note: rationalize returns an unexpanded expression. If the result is assigned to a symbol, evaluating the symbol will expand the result. Use binding to retrieve the unexpanded expression.

f = rationalize(1/a + 1/b + 1/2)
binding(f)

 2 a + a b + 2 b 
╶───────────────╴
      2 a b      

real(z)

Returns the real part of complex z.

real(2 - 3i)

2

rect(z)

Returns complex z in rectangular form.

rect(exp(i x))

cos(x) + i sin(x)

roots(p,x)

Returns the rational roots of a polynomial. Multiple roots are returned as a vector. See also nroots.

p = (x + 1) (x - 2)
roots(p,x)

┌    ┐
│ −1 │
│    │
│ 2  │
└    ┘

If no roots are found then nil is returned. A nil result prints as blank so the following example uses infixform to print nil as a string.

p = x^2 + 1
infixform(roots(p,x))

nil

rotate(u,s,k,...)

Rotates vector u and returns the result. Vector u is required to have 2ⁿ elements where n is an integer from 1 to 15. Arguments s,k,... are a sequence of rotation codes where s is an upper case letter and k is a qubit number 0 to n − 1. Rotations are evaluated from left to right. The available rotations are

Control prefix C, k modifies the next rotation code so that it is a controlled rotation with k as the control qubit. Use two or more prefixes to specify multiple control qubits. For example, C, k, C, j, X, m is a Toffoli rotation. Fourier rotations Q, k and V, k are applied to qubits 0 through k. (Q and V ignore any control prefix.) See also the quantum computing section of the Eigenmath manual.

psi = (1,0,0,0)
rotate(psi,H,0)

┌        ┐
│   1    │
│ ╶────╴ │
│   1/2  │
│  2     │
│        │
│   1    │
│ ╶────╴ │
│   1/2  │
│  2     │
│        │
│   0    │
│        │
│   0    │
└        ┘

run(x)

Run script x where x evaluates to a filename string. Useful for importing function libraries.

run("/Users/heisenberg/EVA2.txt")

For Eigenmath installed from the Mac App Store, run files need to be put in the directory ~/Library/Containers/eigenmath/Data/ and the filename does not require a path.

run("EVA2.txt")

simplify(x)

Returns expression x in a simpler form.

simplify(sin(x)^2 + cos(x)^2)

1

sin(x)

Returns the sine of x.

sin(pi/4)

  1   
╶────╴
  1/2 
 2    

sinh(x)

Returns the hyperbolic sine of x.

circexp(sinh(x))

  1             1        
−╶─╴ exp(−x) + ╶─╴ exp(x)
  2             2        

sqrt(x)

Returns the square root of x.

sqrt(10!)

     1/2
720 7

stop

In a script, it does what it says.

sum(i,j,k,f)

For i equals j through k evaluate f. Returns the sum of all f.

sum(j,1,5,x^j)

 5    4    3    2
x  + x  + x  + x  + x

The original value of i is restored after sum completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of sum.

sum(y)

Returns the sum of components of y.

y = (1,2,3,4)
sum(y)

10

tan(x)

Returns the tangent of x.

tanh(x)

Returns the hyperbolic tangent of x.

circexp(tanh(x))

       1             exp(2 x)   
−╶────────────╴ + ╶────────────╴
  exp(2 x) + 1     exp(2 x) + 1 

taylor(f,x,n,a)

Returns the nth order Taylor series expansion of f(x) at a. If argument a is omitted then zero is used for the expansion point.

taylor(1/(1-x),x,5)

 5    4    3    2        
x  + x  + x  + x  + x + 1

test(a,b,c,d,...)

If argument a is true (nonzero) then b is returned, else if c is true then d is returned, etc. If the number of arguments is odd then the final argument is returned if all else fails. Expressions can include the relational operators =, ==, <, <=, >, >=. Use the not function to test for inequality. (The equality operator == is available for contexts in which = is the assignment operator.)

A = 1
B = 1
test(A=B,"yes","no")

yes

trace

Set trace=1 in a script to print the script as it is evaluated. Useful for debugging.

trace = 1

Note: The contract function is used to obtain the trace of a matrix.

transpose(a,i,j)

Returns the transpose of tensor a with respect to indices i and j. If i and j are omitted then 1 and 2 are used. Hence a matrix can be transposed with a single argument.

A = ((a,b),(c,d))
transpose(A)

┌       ┐
│ a   c │
│       │
│ b   d │
└       ┘

Note: The argument list can be extended for multiple transpose operations. The arguments are evaluated from left to right. For example, transpose(A,1,2,2,3) is equivalent to transpose(transpose(A,1,2),2,3).

tty

Set tty=1 to show results in string format. Set tty=0 to turn off. Can be useful when displayed results exceed window size.

tty = 1
(x + 1)^2

x^2 + 2 x + 1

unit(n)

Returns an n by n identity matrix.

unit(3)

┌           ┐
│ 1   0   0 │
│           │
│ 0   1   0 │
│           │
│ 0   0   1 │
└           ┘

zero(i,j,...)

Returns a null tensor with dimensions i, j, etc. Useful for creating a tensor and then setting the component values.

A = zero(3,3)
for(k,1,3,A[k,k]=k)
A

    ┌           ┐
    │ 1   0   0 │
    │           │
A = │ 0   2   0 │
    │           │
    │ 0   0   3 │
    └           ┘