vectorgebra 2.4.0

A numerical methods tool for python, in python.

Vectorgebra

A numerical methods tool for python, in python.

There are 6 main subclasses; Vector, Matrix, Graph, complex, Infinity, Undefined. 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 2.4.0

Graph class added with basic functionality.

Givens Rotations added to both matrices and vectors.

Jacobi algorithm for solving Ax = b equations is added.

Bug fix on Matrix print method. Now it is dynamic.

A = L + D + U decomposition is added to Matrix class.

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, decimal=True)

Returns dim dimensional vector which has its elements randomly selected as integers within the interval (a, b). If decimal is true, generated contents are decimal objects.

Vectorgebra.Vector.randVfloat(dim, a, b, decimal=True)

Returns dim dimensional vector which has its elements randomly selected as floats within the interval (a, b). If decimal is true, generated contents are decimal objects.

Vectorgebra.Vector.randVbool(dim, decimal=True)

Returns dim dimensional vector which has its elements randomly selected as booleans. If decimal is true, generated contents are decimal objects.

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.outer(v, w)

Returns the outer product of Vectors v and w. Return type is therefore a Matrix.

Vectorgebra.Vector.cumsum()

Returns the cumulative sum.

Vectorgebra.Vector.zero(dim)

Returns dim dimensional zero vector.

Vectorgebra.Vector.one(dim)

Returns dim dimensional all ones vector.

Vectorgebra.Vector.reshape(a, b)

Returns the reshaped matrix.

Vectorgebra.Vector.rotate(i, j, angle, resolution: int = 15)

Rotates the vector, self, around axes "i" and "j" by "angle". "resolution" argument is passed to cos() and sin(). Rotation is done via Givens rotation matrix.

---

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.

"pow" method accepts the parameter "decimal".

Methods are listed below.

Vectorgebra.Matrix.determinant(choice="echelon")

Returns the determinant of the matrix m. Two choices are available currently; echelon, analytic. Default is echelon.

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.conjugate()

Returns the complex conjugate of the self.

Vectorgebra.Matrix.normalize()

Divides self with its determinant.

Vectorgebra.Matrix.hconj()

Returns the Hermitian conjugate of self.

Vectorgebra.Matrix.norm(resolution: int = 15, decimal: bool = True)

Returns the Frobenius norm of self. Utilizes the eigenvalue function. Parameters are directly passed to eigenvalue function.

Vectorgebra.Matrix.inverse(method="iteraitve", resolution=10, lowlimit=0.0000000001, highlimit=100000)

Returns the inverse matrix of self. Returns None if not invertible. method can ben "analytic", "gauss", "neumann" or "iterative". Default is iterative which uses Newton's method for matrix inversion. Resolution is the number of iterations. lowlimit and highlimit are only for gauss method. They control the "resolution" of multiplication and divisions. See the source code for a better inside look.

Neumann will only work for the right conditioned matrices (see here). Neumann only uses resolution parameter.

Vectorgebra.Matrix.identity(dim, decimal=True)

Returns dimxdim dimensional identity matrix. If decimal is true, generated contents are decimal objects.

Vectorgebra.Matrix.zero(dim, decimal=True)

Returns dimxdim dimensional all 0 matrix. If decimal is true, generated contents are decimal objects.

Vectorgebra.Matrix.one(dim, decimal=True)

Returns dimxdim dimensional all 1 matrix. If decimal is true, generated contents are decimal objects.

Vectorgebra.Matrix.randMint(m, n, a, b, decimal=True)

Returns mxn matrix of random integers selected from the interval (a, b). If decimal is true, generated contents are decimal objects.

Vectorgebra.Matrix.randMfloat(m, n, a, b, decimal=True)

Returns mxn matrix of random floats selected from the interval (a, b). If decimal is true, generated contents are decimal objects.

Vectorgebra.Matrix.randMbool(m, n, decimal=True)

Returns mxn matrix of random booleans. If decimal is true, generated contents are decimal objects.

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.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.cholesky()

Applies Cholesky decomposition to self, and returns the L matrix. Applies algorithm is textbook Cholesky–Banachiewicz algorithm.

Vectorgebra.Matrix.get_diagonal()

Returns the diagonal Matrix such that A = L + D + U.

Vectorgebra.Matrix.get_upper()

Returns the upper triangular Matrix such that A = L + D + U.

Vectorgebra.Matrix.get_lower()

Returns the lower triangular Matrix such that A = L + D + U.

Vectorgebra.Matrix.givens(dim, i, j, angle, resolution: int = 15)

Returns the Givens rotation matrix that applies rotation around axes "i"-"j" by "angle". Matrix is dimxdim dimensional. "resolution" is passed to cos() and sin()

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.

Vectorgebra.Matrix.gauss_seidel(b: Vector, initial=None, resolution=15, decimal=True)

Applies Gauss-Seidel method to the equation self * x = b. Returns the resultant x Vector. If "initial" left unchanged, code creates an initial guess by default. This may not converge though. "resolution" is the number of iterations. "decimal" argument is also passed to called functions inside this method. Same for resolution.

Vectorgebra.Matrix.least_squares(b, *args)

Accepts every argument that .inverse() accepts. Solves the equation self * x = b for x. Every argument except b is passed to .inverse().

Vectorgebra.Matrix.jacobi_solve(b, resolution: int = 15)

Solves the equation self * x = b for x via Jacobi algorithm. "resolution" is the number of iterations. Returns the x vector.

---

Vectorgebra.Graph

The class for graphs. These can be constructed via matrices or manually. Can be both directed or undirected.

Constructor accepts 5 arguments; vertices, edges, weights, matrix, directed. "directed" is a bool and False by default. If "matrix" is given, "edges" and "weights" are ignored. If given, "vertices" is not ignored. If left blank, vertices are named numerically. Matrix must be a square, obviously.

Is manually constructed; "vertices" is the list (or tuple) of vertices names. Items can be anything that is hashable. "edges" is a list (or tuple) of length-2 lists (or tuples). Both items must be valid names of vertices, also named in the "vertices" argument. Weights to these edges are passed in-order from "weights" list if is not None. If "weights" is left blank, 1 is assigned as weight to given edges. The adjacency matrix is always generated.

Related data can be accessed through; self.vertices, self.edges, self.weights, self.matrix, self.directed.

Print method is overloaded. Printing is much better than matrices even though it kind of prints the adjacency matrix.

Vectorgebra.Graph.addedge(label, weight=1)

Add an edge with vertex pair "label". Weight is 1 by default. Returns self.

Vectorgebra.Graph.popedge(label)

Pops the edge and returns it defined by "label". If there is more than one, pops the first instance.

Vectorgebra.Graph.addvertex(v)

Adds a vertex named "v". Adjacency matrix is regenerated here.

Vectorgebra.Graph.popvertex(v)

Removes the vertex named "v" and returns it. Removes all edges connected to vertex "v". Adjacency matrix is naturally regenerated.

Vectorgebra.Graph.getdegree(vertex)

Returns the degree of vertex.

Vectorgebra.Graph.getindegree(vertex)

Returns the in-degree of vertex if the graph is directed. Otherwise just returns the undirected degree.

Vectorgebra.Graph.getoutdegree(vertex)

Returns the out-degree of vertex if the graph is directed. Otherwise just returns the undirected degree.

Vectorgebra.Graph.getdegrees()

Returns a dictionary of degrees and vertices. Keys are degrees, values are corresponding vertices' labels.

Vectorgebra.Graph.getweight(label)

Returns the weight og the edge given via label.

Vectorgebra.Graph.isIsomorphic(g, h)

Returns True if g and h are isomorphic, False otherwise. g and h must be Graphs.

Vectorgebra.Graph.isEuler()

Returns True if self is an Euler graph, False otherwise.

---

Constants

Pi, e, log2(e), log2(10), ln(2), sqrt(2), sqrt(pi), sqrt(2 * pi).

Functions

Vectorgebra.Range(low, high, 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)

Returns the absolute value of the argument.

Vectorgebra.sqrt(arg, 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, power: int)

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

Vectorgebra.e(exponent, 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.

Logarithms

There are 4 distinct logarithm functions: log2, ln, log10, log. Each have the arguments x and resolution. "resolution" is the number of iterations and by default is set to 15. "log" function also takes a "base" parameter.

All logarithms are calculated based on "log2" function. "ln" and "log10" use the related constants instead of recalculating the same value over and over again.

Vectorgebra.sigmoid(x, a=1)

Returns the sigmoid functions value at x, where a is the coefficient of x.

Vectorgebra.Sum(f, a, b, step=Decimal(0.01), control: bool=False, limit=Decimal(0.000001))

Returns the sum of f(x) from a to b. step is the step for Range. If control is true, stops the sum when the absolute value of the derivative drops under "limit".

Vectorgebra.mode(data)

Returns the mode of the data. Tuples, lists, vectors and matrices are accepted.

Vectorgebra.mean(data)

Calculates the mean of data. "data" must be a one dimensional iterable.

Vectorgebra.expectation(values, probabilities, moment: int = 1)

"values" and "probabilities" are one dimensional iterables and their lengths must be equal. There is no value checking for the probabilities. If they sum up to more than 1 or have negative values, it is up to the user to check that. "moment" is the power of "values". Returns the expectation value of given data.

Vectorgebra.variance(values, probabilities)

Same constraints as "expectation" apply here. Returns the variance of the given data.

Vectorgebra.sd(values, probabilities)

Same constraints as "variance" apply here. Returns the standard deviation of the given data.

Vectorgebra.maximum(dataset)

Returns the maximum value of dataset. Dataset can be anywhere from tuples to Matrices.

Vectorgebra.minimum(dataset)

Returns the minimum value of dataset. Dataset can be anywhere from tuples to Matrices.

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.

Vectorgebra.binomial(n: int, k: int, p: float)

Calculates the probability according to the binomial distribution. n is the maximum number of events, k is the events that p describes, p is the probability of the "event" happening.

Vectorgebra.geometrical(n: int, p: float)

Calculates the probability according to the geometric distribution. n is the number of total events. p is the probability that the event happens.

Vectorgebra.poisson(k, l)

Calculates the probability according to the Poisson formula. l is the lambda factor. k is the "variable" on the whatever system this function is used to describe.

Vectorgebra.normal(x, resolution: int = 15)

Calculates the normal gaussian formula given x. "resolution" is directly passed to the e() function.

Vectorgebra.gaussian(x, mean, sigma, resolution: int = 15)

Calculates the gaussian given the parameters. "resolution" is directly passed to the e() function.

Vectorgebra.laplace(x, sigma, resolution: int = 15)

Calculates the Laplace distribution given the parameters. "resolution" is directly passed to the e() function.

Vectorgebra.linear_fit(x, y, rate = Decimal(0.01), iterations: int = 15)

Returns the b0 and b1 constants for the linear regression of the given data. x and y must be one dimensional iterables and their lengths must be equal. "rate" is the learning rate. "iterations" is the total number of iterations that this functions going to update the coefficients.

Vectorgebra.general_fit(x, y, rate = Decimal(0.0000002), iterations: int = 15, degree: int = 1)

Calculates the coefficients for at degree polynomial regression. Default rate argument is much much lower because otherwise result easily blows up. Returns the coefficients starting from the zeroth degree as a Vector object.

Internally, x and y sets are converted to Vectors if they were not, so it is faster to initialize them as Vectors.

Vectorgebra.kmeans(dataset, k=2, iterations=15, a = 0, b = 10)

Applies the K-means algorithm on the dataset. "k" is the number of points to assign data clusters. "iterations" is the number of iterations that the algorithm applies. Dataset must be non-empty. Each row of dataset is converted to Vectors internally. Predefining them as such would make the function faster.

Every element of dataset must be of the same type whether they are Vectors or not. Type-checking is based on the first row of the dataset.

Returns a 2-element tuple. First element is a list of Vectors which point to the cluster centers. Second element is a list of Vectors which consist of the initial data. This list has the same length as number of generated cluster centers. Each internal list corresponds to the same indexed center point. So this data is grouped by cluster centers.

Initial guesses are random points whose components are random floats between a and b.

This function does not have decimal support yet.

Trigonometrics

All are of the format Vectorgebra.name(x, 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.

Exits the main loop if the maximum thread count is reached. This can be used as a limiter when given b=Infinity(). However, the thread count limit is most likely 4096.

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.

---

Vectorgebra.Infinity

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 "Undefined". This is a special class that has every mathematical and logical operation overloaded.

Vectorgebra.Undefined

A special class that corresponds to "undefined" in mathematics.

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. Can take in a str argument flagged as "hint".

Vectorgebra.RangeError

Raised when given arguments are out of required range.

Vectorgebra.AmountError

Raised when the amount of arguments in a function is wrong.