vectorgebra 1.5.0

A Python-based math library with linear algebra

Vectorgebra

A numerical methods tool for python

There are 4 main subclasses; Vector, Matrix, complex, Infinity. And also there are functions, constants and exception classes. Each section is explained below.

Project details

pip install vectorgebra

https://pypi.org/project/vectorgebra/

Update notes on 1.5.0

Infinity class added. This class functions as, the infinity. May be positive or negative. All mathematical and logical operators are overloaded. Vector, Matrix and complex classes are currently being updated to include infinities in their function definitions. The process is yet incomplete.

sqrt() function now accepts complex class. Before only option for square roots of complex numbers were complex.sqrt(complex).

Factorials, permutations and combinations added. Future versions will include statistics related functions.

Initialization conditions of vectors are not a mess anymore. They have been properly configured now. Gauss inversion for 20x20 random matrices became 2 milliseconds faster on average by just this change.

Vectorgebra.Vector

Includes basic and some sophisticated operations on vectors.

Addition, multiplication subtraction, division and operations alike are implemented as overloads. Comparison operators compare the length of the vectors. Only exception is == which returns True if and only if all the terms of the vectors are equal.

Methods are listed below.

Vectorgebra.Vector.dot(v)

Returns the dot product of self with v.

Vectorgebra.Vector.append(v)

Appends the argument to the vector. Returns the new vector. Argument can be int, float or Vector.

Vectorgebra.Vector.copy()

Returns the copy of the vector.

Vectorgebra.Vector.pop(ord)

Functions exactly the same as .pop() for the list class. If left blank, pops the last element and returns it. If specified, pops the intended element and returns it.

Vectorgebra.Vector.length()

Returns the length of self.

Vectorgebra.Vector.proj(v)

Projects self onto v and returns the resulting vector. Due to the division included, may result in inaccurate values that should have been zero. However, these values are very close to zero and are of magnitude 10^-17. This situation is important for the method Vector.spanify().

Vectorgebra.Vector.unit()

Returns the unit vector.

Vectorgebra.Vector.spanify(*args)

Applies Gram-Schmidt process to the given list of vectors. Returns the list of resulting vectors. May have inaccuracies explained for Vector.dot() method.

Vectorgebra.Vector.does_span(*args)

Returns True if the given list of vectors span the Rⁿ space where n is the number of vectors. Returns False otherwise. Eliminates the possible error from Vector.spanify() method. Therefore, this method will work just fine regardless the errors from divisions.

Vectorgebra.Vector.randVint(dim , a, b)

Returns dim dimensional vector which has its elements randomly selected as integers within the interval (a, b).

Vectorgebra.Vector.randVfloat(dim, a, b)

Returns dim dimensional vector which has its elements randomly selected as floats within the interval (a, b).

Vectorgebra.Vector.randVbool(dim)

Returns dim dimensional vector which has its elements randomly selected as booleans.

Vectorgebra.Vector.determinant(*args)

Returns the determinant of the matrix which has rows given as vectors in *args. This method is not intended for casual use. It is a background tool for cross product and the determinant method for Matrix class.

Vectorgebra.Vector.cross(*args)

Returns the cross product of the vectors given in *args.

Vectorgebra.Vector.cumsum()

Returns the cumulative sum.

Vectorgebra.Vector.reshape(a, b)

Returns the reshaped matrix.

Footnotes

Rotation is not implemented, because only reasonable way to implement it would be for just 2D and 3D vectors, which is not satisfying.

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Vectorgebra.Matrix

Includes basic operations for matrices.

Basic operations like addition, multiplication subtraction, division are implemented as overloads. Only comparison operator implemented is == which returns true if and only if all the elements of the matrices are equal.

Methods are listed below.

Vectorgebra.Matrix.determinant(m)

Returns the determinant of the matrix m.

Vectorgebra.Matrix.append(arg)

Appends arg as a new row to self. Only accepts vectors as arguments.

Vectorgebra.Matrix.copy()

Returns the copy of the matrix.

Vectorgebra.Matrix.pop(ord)

Functions exactly the same as .pop() in list class. If left blank, pops the last row and returns it. If specified, pops the row in the given order and returns it.

Vectorgebra.Matrix.transpose()

Returns the transpose matrix of self

Vectorgebra.Matrix.inverse(method="iteraitve", resolution=10)

Returns the inverse matrix of self. Returns None if not invertible. method can ben "analytic", "gauss" or "iterative". Default is iterative which uses Newton's method for matrix inversion. Resolution is the number of iterations.

Vectorgebra.Matrix.identity(dim)

Returns dimxdim dimensional identity matrix.

Vectorgebra.Matrix.zero(dim)

Returns dimxdim dimensional all 0 matrix.

Vectorgebra.Matrix.one(dim)

Returns dimxdim dimensional all 1 matrix.

Vectorgebra.Matrix.randMint(m, n, a, b)

Returns mxn matrix of random integers selected from the interval (a, b).

Vectorgebra.Matrix.randMfloat(m, n, a, b)

Returns mxn matrix of random floats selected from the interval (a, b).

Vectorgebra.Matrix.randMbool(m, n)

Returns mxn matrix of random booleans.

Vectorgebra.Matrix.echelon()

Returns reduced row echelon form of self. Also does reorganization on rows and multiplies one of them by -1 every 2 reorganization. This is for the determinant to remain unchanged.

Vectorgebra.Matrix.det_echelon()

Returns determinant of self via .echelon() method. This is faster than the other determinant method, which also loses time during type conversions. But this method does not return the exact value and has a very small error compared to the original determinant. This is due to floating point numbers and can be disregarded for most of the uses.

Vectorgebra.Matrix.fast_inverse()

Underlying algorithm of this inverse is completely based on echelon forms. Sometimes may get some rows wrong, this is due to floating point numbers. Error amount increases as the dimensions of the matrix increases.

Vectorgebra.Matrix.cramer(a, number)

Applies Cramers rule to the equation system represented by the matrix a. number indicates which variable to calculate.

Vectorgebra.Matrix.cumsum()

Returns the cumulative sum.

Vectorgebra.Matrix.reshape(*args)

Returns the reshaped matrix/vector. If the return is a matrix, makes a call to the vectors reshape.

Vectorgebra.Matrix.eigenvalue(resolution: int)

Calculates the eigenvalues of self and returns a list of them. This function cannot calculate complex valued eigenvalues. So if there are any, there will be incorrect numbers in the returned list instead of the complex ones.

The underlying algorithm is QR decomposition and iteration. Resolution is the number of iterations. Default is 10.

Vectorgebra.Matrix.qr()

Applies QR decomposition to self and returns the tuple (Q, R). The algorithm just uses .spanify() from Vector class. If the columns of self do not consist of independent vectors, returns matrices of zeroes for both Q and R. This is to prevent type errors that may have otherwise risen from written code.

Vectorgebra.Matrix.trace()

Returns the trace of self.

Vectorgebra.Matrix.diagonals()

Returns the list of diagonals.

Vectorgebra.Matrix.diagonal_mul()

Returns the multiplication of diagonals.

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Constants

Pi, e and ln(2). Nothing more.

Functions

Vectorgebra.Range(low: int or float, high: int or float, step)

A lazy implementation of range. There is indeed no range. Just a loop with yield statement. Almost as fast as the built-in range.

Vectorgebra.abs(arg: int or float)

Returns the absolute value of the argument.

Vectorgebra.sqrt(arg: int or float, resolution: int = 10)

A square root implementation that uses Newton's method. You may choose the resolution, but any change is not needed there. Pretty much at the same accuracy as the built-in math.sqrt(). Accepts negative numbers too.

Vectorgebra.cumsum(arg: list or float)

Returns the cumulative sum of the iterable.

Vectorgebra.__cumdiv(x: int or float, power: int)

Calculates xⁿ / power!. This is used to calculate Taylor series without data loss (at least minimal data loss).

Vectorgebra.e(exponent: int or float, resolution: int)

Calculates e^exponent. resolution is passed as power to the __cumdiv(). It will then define the maximal power of the power series implementation, therefore is a resolution.

Vectorgebra.factorial(x: int)

Calculates the factorial with recursion. Default argument is 1.

Vectorgebra.permutation(x: int, y: int)

Calculates the y permutations of x elements. Does not utilize the factorial function. Indeed, uses loops to calculate the aimed value more efficiently.

Vectorgebra.combination(x: int, y: int)

Calculates y combinations of x elements. Again, this does not utilize the factorial function.

Vectorgebra.multinomial(n: int, *args)

Calculates the multinomial coefficient with n elements, with partitions described in "args". Does not utilize the factorial.

Trigonometrics

All are of the format Vectorgebra.name(x: int or float, resolution: int). Calculates the value of the named trigonometric function via Taylor series. Again, resolution is passed as power to __cumdiv().

Inverse trigonometrics(arcsin, arccos) do not use the helper function __cumdiv().

Vectorgebra.__find(...)

This is a helper function for Math.solve(). Arguments are the same. Returns the first zero that it finds and saves it to memory.

Vectorgebra.solve(f, low, high, search_step, res)

Finds zeroes of function f. It may not be able to find all zeroes, but is pretty precise when it finds some. If the functions derivative large around its zero, then you should increase resolution to do a better search.

Retrieves found zeroes from the memory, then clears it. Calling multiple instances of this function at the same time will result in errors because of this global memory usage.

This function is optimized for polynomials. It doesn't matter how many zeroes they have since this function utilizes a thread pool. This solver is slow when utilized with Taylor series based functions.

There is an obvious solution to this speed problem though. Just put expanded form as the argument. Not the implicit function form.

Vectorgebra.derivative(f, x, h)

Takes the derivative of f around x with h = h. There is no algorithm here. It just calculates the derivative.

Vectorgebra.integrate(f, a, b, delta)

Calculates the integral of f(x) in the interval (a, b) with the specified delta. Default for delta is 0.01. Uses the midpoint rule.

Vectorgebra.__mul(...)

A helper function to Vectorgebra.matmul(). Threads inside matmul call this function.

Vectorgebra.matmul(m1, m2, max=10)

Threaded matrix multiplication. Its speed is depended on dimensions of matrices. Let it be axb and bxc, (a - b) is proportional to this functions speed. Worst case scenario is square matrices. 44x44 (On CPython) is limit for this function to be faster than the overload version of matrix multiplication.

If b > a, normally this function gets even more slower. But there is a way around. Let it be b > a;

A * B = C

B^T * A^T = C^T

After taking the transposes, we get a > b again. All we have to do is to calculate the matrix C^T instead of C directly then to calculate the transpose of it.

(I didn't add this function to Matrix class because I have more plans on it.)

Vectorgebra.findsol(f, x, resolution)

Calculates a single solution of f with Newton's method. x is the starting guess. resolution is the number of iterations.

Vectorgebra.complex

This is the complex number class. It has + - / * overloaded.

Vectorgebra.complex.conjugate()

Returns the complex conjugate of self.

Vectorgebra.complex.length()

Returns the length of self, with treating it as a vector.

Vectorgebra.complex.unit()

Treats the complex number as a vector and returns the unit vector.

Vectorgebra.complex.sqrt(arg, resolution: int = 200)

Calculates the square root of the complex number arg and returns it again, as a complex number. Resolution argument is only passed to arcsin since it is the only limiting factor for this functions accuracy. Has an average of 1 degree of error as angle. You may still increase the resolution. But reaching less than half a degree of error requires for it to be at least 600.

The used algorithm calculates the unit vector as e^i*x. Then halves the degree, x. Returns the resultant vector at the proper length.

Vectorgebra.complex.range(lowreal, highreal, lowimg, highimg, step1, step2)

Creates a complex number range, ranging from complex(lowreal, lowimg) to complex(highreal, highimg). Steps are 1 by default. Again this is a lazy implementation.

Vectorgebra.complex.inverse()

Returns 1 / self. This is used in division. If divisor is complex, then this function is applied with multiplication to get the result.

Vectorgebra.complex.rotate(angle: int or float)

Rotates self by angle.

Vectorgebra.complex.rotationFactor(angle: int or float)

Returns e^i*angle as a complex number.

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Vectorgebra.Infinity (unstable)

The class of infinities. When initialized, takes one argument describing its sign. If "True", the infinity is positive (which is the default). If "False", the infinity is negative.

Logical and mathematical operations are all overloaded. Operations may return "None". For example infinity - infinity is None. We don't know the preceding expressions. Result may converge or diverge depending on the expressions that infinities arise from. So the operation just returns None. This may cause exceptions. In the future, these cases will return an "Unknown" number class. But this class is not implemented yet.

Exceptions

Vectorgebra.DimensionError

Anything related to dimensions of vectors and matrices. Raised in Vector class when dimensions of operands don't match or 0 is given as a dimension to random vector generating functions.

This error is raised in Matrix class when non-square matrices are passed into inverse calculating functions.

DimensionError(1) has been changed to RangeError, but is still in the code.

Vectorgebra.ArgTypeError

Anything related to types of arguments. There are 8 modes of this exception depending on the conditions. These modes are defined by different combinations of types. For example type "i" is used for errors about arguments that should have been only integers.

Vectorgebra.ArgumentError

Raised when an incorrect amount of arguments is passed into functions.

Vectorgebra.RangeError

Raised when given arguments are out of required range.

Vectorgebra.MathArgError

Raised when argument(s) are of wrong type.

Vectorgebra.MathRangeError

Raised when argument(s) are off range.