Source code for sympy.matrices.matrices

from sympy import Basic, Symbol, Integer, C, S, Dummy, Rational, Add, Pow
from sympy.core.numbers import Zero
from sympy.core.sympify import sympify, converter, SympifyError
from sympy.core.compatibility import is_sequence

from sympy.polys import Poly, roots, cancel
from sympy.simplify import simplify as sympy_simplify
from sympy.utilities.iterables import flatten
from sympy.functions.elementary.miscellaneous import sqrt, Max, Min
from sympy.functions.elementary.complexes import re, Abs
from sympy.printing import sstr

from sympy.core.compatibility import callable, reduce

import random

class MatrixError(Exception):
    pass

class ShapeError(ValueError, MatrixError):
    """Wrong matrix shape"""
    pass

class NonSquareMatrixError(ShapeError):
    pass

def _dims_to_nm(dims):
    """Converts dimensions tuple (or any object with length 1 or 2) or scalar
    in dims to matrix dimensions n and m."""

    try:
        l = len(dims)
    except TypeError:
        dims = (dims,)
        l = 1

    # This will work for nd-array too when they are added to sympy.
    for dim in dims:
        if dim < 0:
            raise ValueError("Matrix dimensions should be non-negative integers.")

    if l == 2:
        n, m = map(int, dims)
    elif l == 1:
        n = m = int(dims[0])
    else:
        raise ValueError("Matrix dimensions should be a two-element tuple of ints or a single int!")

    return n, m

def _iszero(x):
    """Returns True if x is zero."""
    return x.is_zero


class DeferredVector(object):
    def __init__(self,name):
        self.name=name

    def __getitem__(self,i):
        component_name = '%s[%d]'%(self.name,i)
        return Symbol(component_name)

    def __str__(self):
        return sstr(self)

    def __repr__(self):
        return sstr(self)


[docs]class Matrix(object): # Added just for numpy compatibility # TODO: investigate about __array_priority__ __array_priority__ = 10.0 def __init__(self, *args): """ Matrix can be constructed with values or a rule. >>> from sympy import Matrix, I >>> Matrix( ((1,2+I), (3,4)) ) #doctest:+NORMALIZE_WHITESPACE [1, 2 + I] [3, 4] >>> Matrix(2, 2, lambda i,j: (i+1)*j ) #doctest:+NORMALIZE_WHITESPACE [0, 1] [0, 2] """ if len(args) == 3 and callable(args[2]): operation = args[2] self.rows = int(args[0]) self.cols = int(args[1]) self.mat = [] for i in range(self.rows): for j in range(self.cols): self.mat.append(sympify(operation(i, j))) elif len(args)==3 and is_sequence(args[2]): self.rows=args[0] self.cols=args[1] mat = args[2] if len(mat) != len(self): raise ValueError('List length should be equal to rows*columns') self.mat = map(lambda i: sympify(i), mat) elif len(args) == 1: mat = args[0] if isinstance(mat, Matrix): self.rows = mat.rows self.cols = mat.cols self.mat = mat[:] return elif hasattr(mat, "__array__"): # NumPy array or matrix or some other object that implements # __array__. So let's first use this method to get a # numpy.array() and then make a python list out of it. arr = mat.__array__() if len(arr.shape) == 2: self.rows, self.cols = arr.shape[0], arr.shape[1] self.mat = map(lambda i: sympify(i), arr.ravel()) return elif len(arr.shape) == 1: self.rows, self.cols = 1, arr.shape[0] self.mat = [0]*self.cols for i in xrange(len(arr)): self.mat[i] = sympify(arr[i]) return else: raise NotImplementedError("Sympy supports just 1D and 2D matrices") elif not is_sequence(mat, include=Matrix): raise TypeError("Matrix constructor doesn't accept %s as input" % str(type(mat))) mat = [] for row in args[0]: if isinstance(row, Matrix): mat.extend(row.tolist()) else: mat.append(row) self.rows = len(mat) if len(mat) != 0: if not is_sequence(mat[0]): self.cols = 1 self.mat = map(lambda i: sympify(i), mat) return self.cols = len(mat[0]) else: self.cols = 0 self.mat = [] for j in xrange(self.rows): if len(mat[j]) != self.cols: raise ValueError("Input %s inconsistant to form a Matrix." % args) for i in xrange(self.cols): self.mat.append(sympify(mat[j][i])) elif len(args) == 0: # Empty Matrix self.rows = self.cols = 0 self.mat = [] else: raise TypeError("Data type not understood")
[docs] def key2ij(self,key): """Converts key=(4,6) to 4,6 and ensures the key is correct.""" if not (is_sequence(key) and len(key) == 2): raise TypeError("wrong syntax: a[%s]. Use a[i,j] or a[(i,j)]" %repr(key)) i,j=key if not (i>=0 and i<self.rows and j>=0 and j < self.cols): print self.rows, " ", self.cols raise IndexError("Index out of range: a[%s]"%repr(key)) return i,j
[docs] def transpose(self): """ Matrix transposition. >>> from sympy import Matrix, I >>> m=Matrix(((1,2+I),(3,4))) >>> m #doctest: +NORMALIZE_WHITESPACE [1, 2 + I] [3, 4] >>> m.transpose() #doctest: +NORMALIZE_WHITESPACE [ 1, 3] [2 + I, 4] >>> m.T == m.transpose() True """ a = [0]*len(self) for i in xrange(self.cols): a[i*self.rows:(i+1)*self.rows] = self.mat[i::self.cols] return Matrix(self.cols,self.rows,a)
T = property(transpose,None,None,"Matrix transposition.")
[docs] def conjugate(self): """By-element conjugation.""" out = Matrix(self.rows,self.cols, lambda i,j: self[i,j].conjugate()) return out
C = property(conjugate,None,None,"By-element conjugation.") @property
[docs] def H(self): """ Hermite conjugation. >>> from sympy import Matrix, I >>> m=Matrix(((1,2+I),(3,4))) >>> m #doctest: +NORMALIZE_WHITESPACE [1, 2 + I] [3, 4] >>> m.H #doctest: +NORMALIZE_WHITESPACE [ 1, 3] [2 - I, 4] """ out = self.T.C return out
@property
[docs] def D(self): """Dirac conjugation.""" from sympy.physics.matrices import mgamma out = self.H * mgamma(0) return out
def __getitem__(self,key): """ >>> from sympy import Matrix, I >>> m=Matrix(((1,2+I),(3,4))) >>> m #doctest: +NORMALIZE_WHITESPACE [1, 2 + I] [3, 4] >>> m[1,0] 3 >>> m.H[1,0] 2 - I """ if type(key) is tuple: i, j = key if type(i) is slice or type(j) is slice: return self.submatrix(key) else: # a2idx inlined if not type(i) is int: try: i = i.__index__() except AttributeError: raise IndexError("Invalid index a[%r]" % (key,)) # a2idx inlined if not type(j) is int: try: j = j.__index__() except AttributeError: raise IndexError("Invalid index a[%r]" % (key,)) if not (i>=0 and i<self.rows and j>=0 and j < self.cols): raise IndexError("Index out of range: a[%s]" % (key,)) else: return self.mat[i*self.cols + j] else: # row-wise decomposition of matrix if type(key) is slice: return self.mat[key] else: k = a2idx(key) if k is not None: return self.mat[k] raise IndexError("Invalid index: a[%s]" % repr(key)) def __setitem__(self, key, value): """ >>> from sympy import Matrix, I >>> m=Matrix(((1,2+I),(3,4))) >>> m #doctest: +NORMALIZE_WHITESPACE [1, 2 + I] [3, 4] >>> m[1,0]=9 >>> m #doctest: +NORMALIZE_WHITESPACE [1, 2 + I] [9, 4] """ if type(key) is tuple: i, j = key if type(i) is slice or type(j) is slice: if isinstance(value, Matrix): self.copyin_matrix(key, value) return if is_sequence(value): self.copyin_list(key, value) return else: # a2idx inlined if not type(i) is int: try: i = i.__index__() except AttributeError: raise IndexError("Invalid index a[%r]" % (key,)) # a2idx inlined if not type(j) is int: try: j = j.__index__() except AttributeError: raise IndexError("Invalid index a[%r]" % (key,)) if not (i>=0 and i<self.rows and j>=0 and j < self.cols): raise IndexError("Index out of range: a[%s]" % (key,)) else: self.mat[i*self.cols + j] = sympify(value) return else: # row-wise decomposition of matrix if type(key) is slice: raise IndexError("Vector slices not implemented yet.") else: k = a2idx(key) if k is not None: self.mat[k] = sympify(value) return raise IndexError("Invalid index: a[%s]"%repr(key)) def __array__(self): return matrix2numpy(self) def __len__(self): """ Return the number of elements of self. Implemented mainly so bool(Matrix()) == False. """ return self.rows * self.cols
[docs] def tolist(self): """ Return the Matrix converted in a python list. >>> from sympy import Matrix >>> m=Matrix(3, 3, range(9)) >>> m [0, 1, 2] [3, 4, 5] [6, 7, 8] >>> m.tolist() [[0, 1, 2], [3, 4, 5], [6, 7, 8]] """ ret = [0]*self.rows for i in xrange(self.rows): ret[i] = self.mat[i*self.cols:(i+1)*self.cols] return ret
def copyin_matrix(self, key, value): rlo, rhi = self.slice2bounds(key[0], self.rows) clo, chi = self.slice2bounds(key[1], self.cols) if value.rows != rhi - rlo or value.cols != chi - clo: raise ShapeError("The Matrix `value` doesn't have the same dimensions " + "as the in sub-Matrix given by `key`.") for i in range(value.rows): for j in range(value.cols): self[i+rlo, j+clo] = sympify(value[i,j]) def copyin_list(self, key, value): if not is_sequence(value): raise TypeError("`value` must be an ordered iterable, not %s." % type(value)) self.copyin_matrix(key, Matrix(value))
[docs] def hash(self): """Compute a hash every time, because the matrix elements could change.""" return hash(self.__str__() )
@property def shape(self): return (self.rows, self.cols) def __rmul__(self,a): if hasattr(a, "__array__") and a.shape != (): return matrix_multiply(a,self) out = Matrix(self.rows,self.cols,map(lambda i: a*i,self.mat)) return out def expand(self): out = Matrix(self.rows,self.cols,map(lambda i: i.expand(), self.mat)) return out def combine(self): out = Matrix(self.rows,self.cols,map(lambda i: i.combine(),self.mat)) return out def subs(self, *args): out = Matrix(self.rows,self.cols,map(lambda i: i.subs(*args),self.mat)) return out def __sub__(self,a): return self + (-a) def __mul__(self,a): if hasattr(a, "__array__") and a.shape != (): return matrix_multiply(self,a) out = Matrix(self.rows,self.cols,map(lambda i: i*a,self.mat)) return out def __pow__(self, num): if not self.is_square: raise NonSquareMatrixError() if isinstance(num, int) or isinstance(num, Integer): n = int(num) if n < 0: return self.inv() ** -n # A**-2 = (A**-1)**2 a = eye(self.cols) s = self while n: if n%2: a *= s n -= 1 s *= s n //= 2 return a elif isinstance(num, Rational): try: P, D = self.diagonalize() except MatrixError: raise NotImplementedError("Implemented only for diagonalizable matrices") for i in range(D.rows): D[i, i] = D[i, i]**num return P * D * P.inv() else: raise NotImplementedError("Only integer and rational values are supported") def __add__(self,a): return matrix_add(self,a) def __radd__(self,a): return matrix_add(a,self) def __div__(self,a): return self * (S.One/a) def __truediv__(self,a): return self.__div__(a)
[docs] def multiply(self,b): """Returns self*b """ return matrix_multiply(self,b)
[docs] def add(self,b): """Return self+b """ return matrix_add(self,b)
def __neg__(self): return -1*self def __eq__(self, a): if not isinstance(a, (Matrix, Basic)): a = sympify(a) if isinstance(a, Matrix) and self.shape == a.shape: return all(self[i, j] == a[i, j] for i in xrange(self.rows) for j in xrange(self.cols)) else: return False def __ne__(self, a): if not isinstance(a, (Matrix, Basic)): a = sympify(a) if isinstance(a, Matrix) and self.shape == a.shape: return any(self[i, j] != a[i, j] for i in xrange(self.rows) for j in xrange(self.cols)) else: return True def __hash__(self): return super(Matrix, self).__hash__() def _format_str(self, strfunc, rowsep='\n'): # Handle zero dimensions: if self.rows == 0 or self.cols == 0: return '[]' # Build table of string representations of the elements res = [] # Track per-column max lengths for pretty alignment maxlen = [0] * self.cols for i in range(self.rows): res.append([]) for j in range(self.cols): string = strfunc(self[i,j]) res[-1].append(string) maxlen[j] = max(len(string), maxlen[j]) # Patch strings together for i, row in enumerate(res): for j, elem in enumerate(row): # Pad each element up to maxlen so the columns line up row[j] = elem.rjust(maxlen[j]) res[i] = "[" + ", ".join(row) + "]" return rowsep.join(res) def __str__(self): return sstr(self) def __repr__(self): return sstr(self)
[docs] def cholesky(self): """ Returns the Cholesky Decomposition L of a Matrix A such that L * L.T = A A must be a square, symmetric, positive-definite and non-singular matrix >>> from sympy.matrices import Matrix >>> A = Matrix(((25,15,-5),(15,18,0),(-5,0,11))) >>> A.cholesky() [ 5, 0, 0] [ 3, 3, 0] [-1, 1, 3] >>> A.cholesky() * A.cholesky().T [25, 15, -5] [15, 18, 0] [-5, 0, 11] """ if not self.is_square: raise NonSquareMatrixError("Matrix must be square.") if not self.is_symmetric(): raise ValueError("Matrix must be symmetric.") return self._cholesky()
def _cholesky(self): """ Helper function of cholesky. Without the error checks. To be used privately. """ L = zeros((self.rows, self.rows)) for i in xrange(self.rows): for j in xrange(i): L[i, j] = (1 / L[j, j]) * (self[i, j] - sum(L[i, k] * L[j, k] for k in xrange(j))) L[i, i] = (self[i, i] - sum(L[i, k] ** 2 for k in xrange(i))) ** (S(1)/2) return L
[docs] def LDLdecomposition(self): """ Returns the LDL Decomposition (L,D) of matrix A, such that L * D * L.T == A This method eliminates the use of square root. Further this ensures that all the diagonal entries of L are 1. A must be a square, symmetric, positive-definite and non-singular matrix. >>> from sympy.matrices import Matrix, eye >>> A = Matrix(((25,15,-5),(15,18,0),(-5,0,11))) >>> L, D = A.LDLdecomposition() >>> L [ 1, 0, 0] [ 3/5, 1, 0] [-1/5, 1/3, 1] >>> D [25, 0, 0] [ 0, 9, 0] [ 0, 0, 9] >>> L * D * L.T * A.inv() == eye(A.rows) True """ if not self.is_square: raise NonSquareMatrixException("Matrix must be square.") if not self.is_symmetric(): raise ValueError("Matrix must be symmetric.") return self._LDLdecomposition()
def _LDLdecomposition(self): """ Helper function of LDLdecomposition. Without the error checks. To be used privately. """ D = zeros((self.rows, self.rows)) L = eye(self.rows) for i in xrange(self.rows): for j in xrange(i): L[i, j] = (1 / D[j, j]) * (self[i, j] - sum( L[i, k] * L[j, k] * D[k, k] for k in xrange(j))) D[i, i] = self[i, i] - sum(L[i, k]**2 * D[k, k] for k in xrange(i)) return L, D
[docs] def lower_triangular_solve(self, rhs): """ Solves Ax = B, where A is a lower triangular matrix. """ if not self.is_square: raise NonSquareMatrixException("Matrix must be square.") if rhs.rows != self.rows: raise ShapeError("Matrices size mismatch.") if not self.is_lower(): raise ValueError("Matrix must be lower triangular.") return self._lower_triangular_solve(rhs)
def _lower_triangular_solve(self, rhs): """ Helper function of function lower_triangular_solve. Without the error checks. To be used privately. """ X = zeros((self.rows, 1)) for i in xrange(self.rows): if self[i, i] == 0: raise TypeError("Matrix must be non-singular.") X[i, 0] = (rhs[i, 0] - sum(self[i, k] * X[k, 0] for k in xrange(i))) / self[i, i] return X
[docs] def upper_triangular_solve(self, rhs): """ Solves Ax = B, where A is an upper triangular matrix. """ if not self.is_square: raise NonSquareMatrixException("Matrix must be square.") if rhs.rows != self.rows: raise TypeError("Matrix size mismatch.") if not self.is_upper(): raise TypeError("Matrix is not upper triangular.") return self._upper_triangular_solve(rhs)
def _upper_triangular_solve(self, rhs): """ Helper function of function upper_triangular_solve. Without the error checks, to be used privately. """ X = zeros((self.rows, 1)) for i in reversed(xrange(self.rows)): if self[i, i] == 0: raise ValueError("Matrix must be non-singular.") X[i, 0] = (rhs[i, 0] - sum(self[i, k] * X[k, 0] for k in xrange(i+1, self.rows))) / self[i, i] return X
[docs] def cholesky_solve(self, rhs): """ Solves Ax = B using Cholesky decomposition, for a general square non-singular matrix. For a non-square matrix with rows > cols, the least squares solution is returned. """ if self.is_symmetric(): L = self._cholesky() elif self.rows >= self.cols: L = (self.T * self)._cholesky() rhs = self.T * rhs else: raise NotImplementedError("Under-determined System.") Y = L._lower_triangular_solve(rhs) return (L.T)._upper_triangular_solve(Y)
[docs] def diagonal_solve(self, rhs): """ Solves Ax = B efficiently, where A is a diagonal Matrix, with non-zero diagonal entries. """ if not self.is_diagonal: raise TypeError("Matrix should be diagonal") if rhs.rows != self.rows: raise TypeError("Size mis-match") return self._diagonal_solve(rhs)
def _diagonal_solve(self, rhs): """ Helper function of function diagonal_solve, without the error checks, to be used privately. """ return Matrix(rhs.rows, 1, lambda i, j: rhs[i, 0] / self[i, i])
[docs] def LDLsolve(self, rhs): """ Solves Ax = B using LDL decomposition, for a general square and non-singular matrix. For a non-square matrix with rows > cols, the least squares solution is returned. """ if self.is_symmetric(): L, D = self.LDLdecomposition() elif self.rows >= self.cols: L, D = (self.T * self).LDLdecomposition() rhs = self.T * rhs else: raise NotImplementedError("Under-determined System.") Y = L._lower_triangular_solve(rhs) Z = D._diagonal_solve(Y) return (L.T)._upper_triangular_solve(Z)
[docs] def inv(self, method="GE", iszerofunc=_iszero, try_block_diag=False): """ Calculates the matrix inverse. According to the "method" parameter, it calls the appropriate method: GE .... inverse_GE() LU .... inverse_LU() ADJ ... inverse_ADJ() According to the "try_block_diag" parameter, it will try to form block diagonal matrices using the method get_diag_blocks(), invert these individually, and then reconstruct the full inverse matrix. Note, the GE and LU methods may require the matrix to be simplified before it is inverted in order to properly detect zeros during pivoting. In difficult cases a custom zero detection function can be provided by setting the iszerosfunc argument to a function that should return True if its argument is zero. """ if not self.is_square: raise NonSquareMatrixError() if try_block_diag: blocks = self.get_diag_blocks() r = [] for block in blocks: r.append(block.inv(method=method, iszerofunc=iszerofunc)) return diag(*r) if method == "GE": return self.inverse_GE(iszerofunc=iszerofunc) elif method == "LU": return self.inverse_LU(iszerofunc=iszerofunc) elif method == "ADJ": return self.inverse_ADJ() else: raise ValueError("Inversion method unrecognized")
def __mathml__(self): mml = "" for i in range(self.rows): mml += "<matrixrow>" for j in range(self.cols): mml += self[i,j].__mathml__() mml += "</matrixrow>" return "<matrix>" + mml + "</matrix>"
[docs] def row(self, i, f): """ Elementary row operation using functor >>> from sympy import ones >>> I = ones(3) >>> I.row(1,lambda i,j: i*3) >>> I [1, 1, 1] [3, 3, 3] [1, 1, 1] """ for j in range(0, self.cols): self[i, j] = f(self[i, j], j)
[docs] def col(self, j, f): """ Elementary column operation using functor >>> from sympy import ones >>> I = ones(3) >>> I.col(0,lambda i,j: i*3) >>> I [3, 1, 1] [3, 1, 1] [3, 1, 1] """ for i in range(0, self.rows): self[i, j] = f(self[i, j], i)
def row_swap(self, i, j): for k in range(0, self.cols): self[i, k], self[j, k] = self[j, k], self[i, k] def col_swap(self, i, j): for k in range(0, self.rows): self[k, i], self[k, j] = self[k, j], self[k, i] def row_del(self, i): self.mat = self.mat[:i*self.cols] + self.mat[(i+1)*self.cols:] self.rows -= 1
[docs] def col_del(self, i): """ >>> import sympy >>> M = sympy.matrices.eye(3) >>> M.col_del(1) >>> M #doctest: +NORMALIZE_WHITESPACE [1, 0] [0, 0] [0, 1] """ for j in range(self.rows-1, -1, -1): del self.mat[i+j*self.cols] self.cols -= 1
[docs] def row_join(self, rhs): """ Concatenates two matrices along self's last and rhs's first column >>> from sympy import Matrix >>> M = Matrix(3,3,lambda i,j: i+j) >>> V = Matrix(3,1,lambda i,j: 3+i+j) >>> M.row_join(V) [0, 1, 2, 3] [1, 2, 3, 4] [2, 3, 4, 5] """ if self.rows != rhs.rows: raise ShapeError("`self` and `rhs` must have the same number of rows.") newmat = self.zeros((self.rows, self.cols + rhs.cols)) newmat[:,:self.cols] = self[:,:] newmat[:,self.cols:] = rhs return newmat
[docs] def col_join(self, bott): """ Concatenates two matrices along self's last and bott's first row >>> from sympy import Matrix >>> M = Matrix(3,3,lambda i,j: i+j) >>> V = Matrix(1,3,lambda i,j: 3+i+j) >>> M.col_join(V) [0, 1, 2] [1, 2, 3] [2, 3, 4] [3, 4, 5] """ if self.cols != bott.cols: raise ShapeError("`self` and `bott` must have the same number of columns.") newmat = self.zeros((self.rows+bott.rows, self.cols)) newmat[:self.rows,:] = self[:,:] newmat[self.rows:,:] = bott return newmat
[docs] def row_insert(self, pos, mti): """ >>> from sympy import Matrix, zeros >>> M = Matrix(3,3,lambda i,j: i+j) >>> M [0, 1, 2] [1, 2, 3] [2, 3, 4] >>> V = zeros((1, 3)) >>> V [0, 0, 0] >>> M.row_insert(1,V) [0, 1, 2] [0, 0, 0] [1, 2, 3] [2, 3, 4] """ if pos is 0: return mti.col_join(self) if self.cols != mti.cols: raise ShapeError("`self` and `mti` must have the same number of columns.") newmat = self.zeros((self.rows + mti.rows, self.cols)) newmat[:pos,:] = self[:pos,:] newmat[pos:pos+mti.rows,:] = mti[:,:] newmat[pos+mti.rows:,:] = self[pos:,:] return newmat
[docs] def col_insert(self, pos, mti): """ >>> from sympy import Matrix, zeros >>> M = Matrix(3,3,lambda i,j: i+j) >>> M [0, 1, 2] [1, 2, 3] [2, 3, 4] >>> V = zeros((3, 1)) >>> V [0] [0] [0] >>> M.col_insert(1,V) [0, 0, 1, 2] [1, 0, 2, 3] [2, 0, 3, 4] """ if pos is 0: return mti.row_join(self) if self.rows != mti.rows: raise ShapeError("self and mti must have the same number of rows.") newmat = self.zeros((self.rows, self.cols + mti.cols)) newmat[:,:pos] = self[:,:pos] newmat[:,pos:pos+mti.cols] = mti[:,:] newmat[:,pos+mti.cols:] = self[:,pos:] return newmat
def trace(self): if not self.is_square: raise NonSquareMatrixError() trace = 0 for i in range(self.cols): trace += self[i,i] return trace
[docs] def submatrix(self, keys): """ >>> from sympy import Matrix >>> m = Matrix(4,4,lambda i,j: i+j) >>> m #doctest: +NORMALIZE_WHITESPACE [0, 1, 2, 3] [1, 2, 3, 4] [2, 3, 4, 5] [3, 4, 5, 6] >>> m[0:1, 1] #doctest: +NORMALIZE_WHITESPACE [1] >>> m[0:2, 0:1] #doctest: +NORMALIZE_WHITESPACE [0] [1] >>> m[2:4, 2:4] #doctest: +NORMALIZE_WHITESPACE [4, 5] [5, 6] """ if not isinstance(keys[0], slice) and not isinstance(keys[1], slice): raise TypeError("At least one element of `keys` must be a slice object.") rlo, rhi = self.slice2bounds(keys[0], self.rows) clo, chi = self.slice2bounds(keys[1], self.cols) if not ( 0<=rlo<=rhi and 0<=clo<=chi ): raise IndexError("Slice indices out of range: a[%s]"%repr(keys)) outLines, outCols = rhi-rlo, chi-clo outMat = [0]*outLines*outCols for i in xrange(outLines): outMat[i*outCols:(i+1)*outCols] = self.mat[(i+rlo)*self.cols+clo:(i+rlo)*self.cols+chi] return Matrix(outLines,outCols,outMat)
[docs] def extract(self, rowsList, colsList): """ Extract a submatrix by specifying a list of rows and columns Examples: >>> from sympy import Matrix >>> m = Matrix(4, 3, lambda i, j: i*3 + j) >>> m #doctest: +NORMALIZE_WHITESPACE [0, 1, 2] [3, 4, 5] [6, 7, 8] [9, 10, 11] >>> m.extract([0,1,3],[0,1]) #doctest: +NORMALIZE_WHITESPACE [0, 1] [3, 4] [9, 10] See also: .submatrix() """ cols = self.cols rows = self.rows mat = self.mat if not all(i < rows for i in rowsList): raise IndexError("Row indices out of range") if not all(j < cols for j in colsList): raise IndexError("Column indices out of range") return Matrix(len(rowsList), len(colsList), lambda i,j: mat[rowsList[i]*cols + colsList[j]])
[docs] def slice2bounds(self, key, defmax): """ Takes slice or number and returns (min,max) for iteration Takes a default maxval to deal with the slice ':' which is (none, none) """ if isinstance(key, slice): lo, hi = 0, defmax if key.start is not None: if key.start >= 0: lo = key.start else: lo = defmax+key.start if key.stop is not None: if key.stop >= 0: hi = key.stop else: hi = defmax+key.stop return lo, hi elif isinstance(key, int): if key >= 0: return key, key+1 else: return defmax+key, defmax+key+1 else: raise IndexError("Improper index type")
[docs] def applyfunc(self, f): """ >>> from sympy import Matrix >>> m = Matrix(2,2,lambda i,j: i*2+j) >>> m #doctest: +NORMALIZE_WHITESPACE [0, 1] [2, 3] >>> m.applyfunc(lambda i: 2*i) #doctest: +NORMALIZE_WHITESPACE [0, 2] [4, 6] """ if not callable(f): raise TypeError("`f` must be callable.") out = Matrix(self.rows,self.cols,map(f,self.mat)) return out
def evalf(self, prec=None, **options): if prec is None: return self.applyfunc(lambda i: i.evalf(**options)) else: return self.applyfunc(lambda i: i.evalf(prec, **options))
[docs] def reshape(self, _rows, _cols): """ >>> from sympy import Matrix >>> m = Matrix(2,3,lambda i,j: 1) >>> m #doctest: +NORMALIZE_WHITESPACE [1, 1, 1] [1, 1, 1] >>> m.reshape(1,6) #doctest: +NORMALIZE_WHITESPACE [1, 1, 1, 1, 1, 1] >>> m.reshape(3,2) #doctest: +NORMALIZE_WHITESPACE [1, 1] [1, 1] [1, 1] """ if len(self) != _rows*_cols: print "Invalid reshape parameters %d %d" % (_rows, _cols) return Matrix(_rows, _cols, lambda i,j: self.mat[i*_cols + j])
[docs] def print_nonzero (self, symb="X"): """ Shows location of non-zero entries for fast shape lookup :: >>> from sympy import Matrix, matrices >>> m = Matrix(2,3,lambda i,j: i*3+j) >>> m #doctest: +NORMALIZE_WHITESPACE [0, 1, 2] [3, 4, 5] >>> m.print_nonzero() #doctest: +NORMALIZE_WHITESPACE [ XX] [XXX] >>> m = matrices.eye(4) >>> m.print_nonzero("x") #doctest: +NORMALIZE_WHITESPACE [x ] [ x ] [ x ] [ x] """ s = "" for i in range(self.rows): s += "[" for j in range(self.cols): if self[i,j] == 0: s += " " else: s += symb + "" s += "]\n" print s
[docs] def LUsolve(self, rhs, iszerofunc=_iszero): """ Solve the linear system Ax = b for x. self is the coefficient matrix A and rhs is the right side b. This is for symbolic matrices, for real or complex ones use sympy.mpmath.lu_solve or sympy.mpmath.qr_solve. """ if rhs.rows != self.rows: raise ShapeError("`self` and `rhs` must have the same number of rows.") A, perm = self.LUdecomposition_Simple(iszerofunc=_iszero) n = self.rows b = rhs.permuteFwd(perm) # forward substitution, all diag entries are scaled to 1 for i in range(n): for j in range(i): b.row(i, lambda x,k: x - b[j,k]*A[i,j]) # backward substitution for i in range(n-1,-1,-1): for j in range(i+1, n): b.row(i, lambda x,k: x - b[j,k]*A[i,j]) b.row(i, lambda x,k: x / A[i,i]) return b
[docs] def LUdecomposition(self, iszerofunc=_iszero): """ Returns the decomposition LU and the row swaps p. Example: >>> from sympy import Matrix >>> a = Matrix([[4, 3], [6, 3]]) >>> L, U, _ = a.LUdecomposition() >>> L [ 1, 0] [3/2, 1] >>> U [4, 3] [0, -3/2] """ combined, p = self.LUdecomposition_Simple(iszerofunc=_iszero) L = self.zeros(self.rows) U = self.zeros(self.rows) for i in range(self.rows): for j in range(self.rows): if i > j: L[i,j] = combined[i,j] else: if i == j: L[i,i] = 1 U[i,j] = combined[i,j] return L, U, p
[docs] def LUdecomposition_Simple(self, iszerofunc=_iszero): """ Returns A comprised of L,U (L's diag entries are 1) and p which is the list of the row swaps (in order). """ if not self.is_square: raise NonSquareMatrixError() n = self.rows A = self[:,:] p = [] # factorization for j in range(n): for i in range(j): for k in range(i): A[i,j] = A[i,j] - A[i,k]*A[k,j] pivot = -1 for i in range(j,n): for k in range(j): A[i,j] = A[i,j] - A[i,k]*A[k,j] # find the first non-zero pivot, includes any expression if pivot == -1 and not iszerofunc(A[i,j]): pivot = i if pivot < 0: # this result is based on iszerofunc's analysis of the possible pivots, so even though # the element may not be strictly zero, the supplied iszerofunc's evaluation gave True raise ValueError("No nonzero pivot found; inversion failed.") if pivot != j: # row must be swapped A.row_swap(pivot,j) p.append([pivot,j]) scale = 1 / A[j,j] for i in range(j+1,n): A[i,j] = A[i,j] * scale return A, p
[docs] def LUdecompositionFF(self): """ Compute a fraction-free LU decomposition. Returns 4 matrices P, L, D, U such that PA = L D**-1 U. If the elements of the matrix belong to some integral domain I, then all elements of L, D and U are guaranteed to belong to I. **Reference** - W. Zhou & D.J. Jeffrey, "Fraction-free matrix factors: new forms for LU and QR factors". Frontiers in Computer Science in China, Vol 2, no. 1, pp. 67-80, 2008. """ n, m = self.rows, self.cols U, L, P = self[:,:], eye(n), eye(n) DD = zeros(n) # store it smarter since it's just diagonal oldpivot = 1 for k in range(n-1): if U[k,k] == 0: for kpivot in range(k+1, n): if U[kpivot, k] != 0: break else: raise ValueError("Matrix is not full rank") U[k, k:], U[kpivot, k:] = U[kpivot, k:], U[k, k:] L[k, :k], L[kpivot, :k] = L[kpivot, :k], L[k, :k] P[k, :], P[kpivot, :] = P[kpivot, :], P[k, :] L[k,k] = Ukk = U[k,k] DD[k,k] = oldpivot * Ukk for i in range(k+1, n): L[i,k] = Uik = U[i,k] for j in range(k+1, m): U[i,j] = (Ukk * U[i,j] - U[k,j]*Uik) / oldpivot U[i,k] = 0 oldpivot = Ukk DD[n-1,n-1] = oldpivot return P, L, DD, U
def cofactorMatrix(self, method="berkowitz"): out = Matrix(self.rows, self.cols, lambda i,j: self.cofactor(i, j, method)) return out def minorEntry(self, i, j, method="berkowitz"): if not 0 <= i < self.rows or not 0 <= j < self.cols: raise ValueError("`i` and `j` must satisfy 0 <= i < `self.rows` " + "(%d)" % self.rows + "and 0 <= j < `self.cols` (%d)." % self.cols) return self.minorMatrix(i,j).det(method) def minorMatrix(self, i, j): if not 0 <= i < self.rows or not 0 <= j < self.cols: raise ValueError("`i` and `j` must satisfy 0 <= i < `self.rows` " + "(%d)" % self.rows + "and 0 <= j < `self.cols` (%d)." % self.cols) return self.delRowCol(i,j) def cofactor(self, i, j, method="berkowitz"): if (i+j) % 2 == 0: return self.minorEntry(i, j, method) else: return -1 * self.minorEntry(i, j, method)
[docs] def jacobian(self, X): """ Calculates the Jacobian matrix (derivative of a vectorial function). *self* A vector of expressions representing functions f_i(x_1, ..., x_n). *X* The set of x_i's in order, it can be a list or a Matrix Both self and X can be a row or a column matrix in any order (jacobian() should always work). Examples:: >>> from sympy import sin, cos, Matrix >>> from sympy.abc import rho, phi >>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2]) >>> Y = Matrix([rho, phi]) >>> X.jacobian(Y) [cos(phi), -rho*sin(phi)] [sin(phi), rho*cos(phi)] [ 2*rho, 0] >>> X = Matrix([rho*cos(phi), rho*sin(phi)]) >>> X.jacobian(Y) [cos(phi), -rho*sin(phi)] [sin(phi), rho*cos(phi)] """ if not isinstance(X, Matrix): X = Matrix(X) # Both X and self can be a row or a column matrix, so we need to make # sure all valid combinations work, but everything else fails: if self.shape[0] == 1: m = self.shape[1] elif self.shape[1] == 1: m = self.shape[0] else: raise TypeError("self must be a row or a column matrix") if X.shape[0] == 1: n = X.shape[1] elif X.shape[1] == 1: n = X.shape[0] else: raise TypeError("X must be a row or a column matrix") # m is the number of functions and n is the number of variables # computing the Jacobian is now easy: return Matrix(m, n, lambda j, i: self[j].diff(X[i]))
[docs] def QRdecomposition(self): """ Return Q,R where A = Q*R, Q is orthogonal and R is upper triangular. Examples This is the example from wikipedia:: >>> from sympy import Matrix, eye >>> A = Matrix([[12,-51,4],[6,167,-68],[-4,24,-41]]) >>> Q, R = A.QRdecomposition() >>> Q [ 6/7, -69/175, -58/175] [ 3/7, 158/175, 6/175] [-2/7, 6/35, -33/35] >>> R [14, 21, -14] [ 0, 175, -70] [ 0, 0, 35] >>> A == Q*R True QR factorization of an identity matrix >>> A = Matrix([[1,0,0],[0,1,0],[0,0,1]]) >>> Q, R = A.QRdecomposition() >>> Q [1, 0, 0] [0, 1, 0] [0, 0, 1] >>> R [1, 0, 0] [0, 1, 0] [0, 0, 1] """ if not self.rows >= self.cols: raise MatrixError("The number of rows must be greater than columns") n = self.rows m = self.cols rank = n row_reduced = self.rref()[0] for i in range(row_reduced.rows): if Matrix(row_reduced[i*m:(i+1)*m]).norm() == 0: rank -= 1 if not rank == self.cols: raise MatrixError("The rank of the matrix must match the columns") Q, R = self.zeros((n, m)), self.zeros(m) for j in range(m): # for each column vector tmp = self[:,j] # take original v for i in range(j): # subtract the project of self on new vector tmp -= Q[:,i] * self[:,j].dot(Q[:,i]) tmp.expand() # normalize it R[j,j] = tmp.norm() Q[:,j] = tmp / R[j,j] if Q[:,j].norm() != 1: raise NotImplementedError("Could not normalize the vector %d." % j) for i in range(j): R[i,j] = Q[:,i].dot(self[:,j]) return Q,R
[docs] def QRsolve(self, b): """ Solve the linear system 'Ax = b'. 'self' is the matrix 'A', the method argument is the vector 'b'. The method returns the solution vector 'x'. If 'b' is a matrix, the system is solved for each column of 'b' and the return value is a matrix of the same shape as 'b'. This method is slower (approximately by a factor of 2) but more stable for floating-point arithmetic than the LUsolve method. However, LUsolve usually uses an exact arithmetic, so you don't need to use QRsolve. This is mainly for educational purposes and symbolic matrices, for real (or complex) matrices use sympy.mpmath.qr_solve. """ Q, R = self.QRdecomposition() y = Q.T * b # back substitution to solve R*x = y: # We build up the result "backwards" in the vector 'x' and reverse it # only in the end. x = [] n = R.rows for j in range(n-1, -1, -1): tmp = y[j,:] for k in range(j+1, n): tmp -= R[j,k] * x[n-1-k] x.append(tmp/R[j,j]) return Matrix([row.mat for row in reversed(x)]) # Utility functions
[docs] def simplify(self, simplify=sympy_simplify, ratio=1.7): """Simplify the elements of a matrix in place. If (result length)/(input length) > ratio, then input is returned unmodified. If 'ratio=oo', then simplify() is applied anyway. See also simplify(). """ for i in xrange(len(self.mat)): self.mat[i] = simplify(self.mat[i], ratio=ratio) #def evaluate(self): # no more eval() so should be removed # for i in range(self.rows): # for j in range(self.cols): # self[i,j] = self[i,j].eval()
def cross(self, b): if not is_sequence(b, include=Matrix): raise TypeError("`b` must be an ordered iterable or Matrix, not %s." % type(b)) if not (self.rows == 1 and self.cols == 3 or \ self.rows == 3 and self.cols == 1 ) and \ (b.rows == 1 and b.cols == 3 or \ b.rows == 3 and b.cols == 1): raise ShapeError("Dimensions incorrect for cross product.") else: return Matrix(1,3,((self[1]*b[2] - self[2]*b[1]), (self[2]*b[0] - self[0]*b[2]), (self[0]*b[1] - self[1]*b[0]))) def dot(self, b): if not is_sequence(b, include=Matrix): raise TypeError("`b` must be an ordered iterable or Matrix, not %s." % type(b)) m = len(b) if len(self) != m: raise ShapeError("Dimensions incorrect for dot product.") prod = 0 for i in range(m): prod += self[i] * b[i] return prod
[docs] def multiply_elementwise(self, b): """Return the Hadamard product (elementwise product) of A and B >>> import sympy >>> A = sympy.Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = sympy.Matrix([[1, 10, 100], [100, 10, 1]]) >>> print A.multiply_elementwise(B) [ 0, 10, 200] [300, 40, 5] """ return matrix_multiply_elementwise(self, b)
[docs] def norm(self, ord=None): """Return the Norm of a Matrix or Vector. In the simplest case this is the geometric size of the vector Other norms can be specified by the ord parameter ===== ============================ ========================== ord norm for matrices norm for vectors ===== ============================ ========================== None Frobenius norm 2-norm 'fro' Frobenius norm - does not exist inf -- max(abs(x)) -inf -- min(abs(x)) 1 -- as below -1 -- as below 2 2-norm (largest sing. value) as below -2 smallest singular value as below other - does not exist sum(abs(x)**ord)**(1./ord) ===== ============================ ========================== >>> from sympy import Matrix, var, trigsimp, cos, sin >>> x = var('x', real=True) >>> v = Matrix([cos(x), sin(x)]) >>> print trigsimp( v.norm() ) 1 >>> print v.norm(10) (sin(x)**10 + cos(x)**10)**(1/10) >>> A = Matrix([[1,1], [1,1]]) >>> print A.norm(2)# Spectral norm (max of |Ax|/|x| under 2-vector-norm) 2 >>> print A.norm(-2) # Inverse spectral norm (smallest singular value) 0 >>> print A.norm() # Frobenius Norm 2 """ # Row or Column Vector Norms if self.rows == 1 or self.cols == 1: if ord == 2 or ord == None: # Common case sqrt(<x,x>) return Add(*(abs(i)**2 for i in self.mat))**S.Half elif ord == 1: # sum(abs(x)) return Add(*(abs(i) for i in self.mat)) elif ord == S.Infinity: # max(abs(x)) return Max(*self.applyfunc(abs)) elif ord == S.NegativeInfinity: # min(abs(x)) return Min(*self.applyfunc(abs)) # Otherwise generalize the 2-norm, Sum(x_i**ord)**(1/ord) # Note that while useful this is not mathematically a norm try: return Pow( Add(*(abs(i)**ord for i in self.mat)), S(1)/ord ) except: raise ValueError("Expected order to be Number, Symbol, oo") # Matrix Norms else: if ord == 2: # Spectral Norm # Maximum singular value return Max(*self.singular_values()) elif ord == -2: # Minimum singular value return Min(*self.singular_values()) elif (ord == None or isinstance(ord,str) and ord.lower() in ['f', 'fro', 'frobenius', 'vector']): # Reshape as vector and send back to norm function return self.vec().norm(ord=2) else: raise NotImplementedError("Matrix Norms under development")
def normalized(self): if self.rows != 1 and self.cols != 1: raise ShapeError("A Matrix must be a vector to normalize.") norm = self.norm() out = self.applyfunc(lambda i: i / norm) return out
[docs] def project(self, v): """Project onto v.""" return v * (self.dot(v) / v.dot(v))
def permuteBkwd(self, perm): copy = self[:,:] for i in range(len(perm)-1, -1, -1): copy.row_swap(perm[i][0], perm[i][1]) return copy def permuteFwd(self, perm): copy = self[:,:] for i in range(len(perm)): copy.row_swap(perm[i][0], perm[i][1]) return copy def delRowCol(self, i, j): # used only for cofactors, makes a copy M = self[:,:] M.row_del(i) M.col_del(j) return M
[docs] def exp(self): """ Returns the exponent of a matrix """ if not self.is_square: raise NonSquareMatrixError("Exponentiation is valid only for square matrices") try: U, D = self.diagonalize() except MatrixError: raise NotImplementedError("Exponentiation is implemented only for diagonalizable matrices") for i in xrange(0, D.rows): D[i, i] = C.exp(D[i, i]) return U * D * U.inv()
[docs] def zeros(self, dims): """Returns a dims = (d1,d2) matrix of zeros.""" n, m = _dims_to_nm( dims ) return Matrix(n,m,[S.Zero]*n*m)
[docs] def eye(self, n): """Returns the identity matrix of size n.""" tmp = self.zeros(n) for i in range(tmp.rows): tmp[i,i] = S.One return tmp
@property def is_square(self): return self.rows == self.cols
[docs] def is_nilpotent(self): """ Checks if a matrix is nilpotent. A matrix B is nilpotent if for some integer k, B**k is a zero matrix. Example: >>> from sympy import Matrix >>> a = Matrix([[0,0,0],[1,0,0],[1,1,0]]) >>> a.is_nilpotent() True >>> a = Matrix([[1,0,1],[1,0,0],[1,1,0]]) >>> a.is_nilpotent() False """ if not self.is_square: raise NonSquareMatrixError("Nilpotency is valid only for square matrices") x = Dummy('x') if self.charpoly(x).args[0] == x**self.rows: return True return False
[docs] def is_upper(self): """ Check if matrix is an upper triangular matrix. Example: >>> from sympy import Matrix >>> m = Matrix(2,2,[1, 0, 0, 1]) >>> m [1, 0] [0, 1] >>> m.is_upper() True >>> m = Matrix(3,3,[5, 1, 9, 0, 4 , 6, 0, 0, 5]) >>> m [5, 1, 9] [0, 4, 6] [0, 0, 5] >>> m.is_upper() True >>> m = Matrix(2,3,[4, 2, 5, 6, 1, 1]) >>> m [4, 2, 5] [6, 1, 1] >>> m.is_upper() False """ for i in xrange(1, self.rows): for j in xrange(0, i): if self[i,j] != 0: return False return True
[docs] def is_lower(self): """ Check if matrix is a lower triangular matrix. Example: >>> from sympy import Matrix >>> m = Matrix(2,2,[1, 0, 0, 1]) >>> m [1, 0] [0, 1] >>> m.is_lower() True >>> m = Matrix(3,3,[2, 0, 0, 1, 4 , 0, 6, 6, 5]) >>> m [2, 0, 0] [1, 4, 0] [6, 6, 5] >>> m.is_lower() True >>> from sympy.abc import x, y >>> m = Matrix(2,2,[x**2 + y, y**2 + x, 0, x + y]) >>> m [x**2 + y, x + y**2] [ 0, x + y] >>> m.is_lower() False """ for i in xrange(0, self.rows): for j in xrange(i+1, self.cols): if self[i, j] != 0: return False return True
[docs] def is_upper_hessenberg(self): """ Checks if the matrix is the upper hessenberg form. The upper hessenberg matrix has zero entries below the first subdiagonal. Example: >>> from sympy.matrices import Matrix >>> a = Matrix([[1,4,2,3],[3,4,1,7],[0,2,3,4],[0,0,1,3]]) >>> a [1, 4, 2, 3] [3, 4, 1, 7] [0, 2, 3, 4] [0, 0, 1, 3] >>> a.is_upper_hessenberg() True """ for i in xrange(2, self.rows): for j in xrange(0, i - 1): if self[i,j] != 0: return False return True
[docs] def is_lower_hessenberg(self): r""" Checks if the matrix is in the lower hessenberg form. The lower hessenberg matrix has zero entries above the first superdiagonal. Example: >>> from sympy.matrices import Matrix >>> a = Matrix([[1,2,0,0],[5,2,3,0],[3,4,3,7],[5,6,1,1]]) >>> a [1, 2, 0, 0] [5, 2, 3, 0] [3, 4, 3, 7] [5, 6, 1, 1] >>> a.is_lower_hessenberg() True """ for i in xrange(0, self.rows): for j in xrange(i + 2, self.cols): if self[i, j] != 0: return False return True
def is_symbolic(self): for element in self.mat: if element.has(Symbol): return True return False
[docs] def is_symmetric(self, simplify=True): """ Check if matrix is symmetric matrix, that is square matrix and is equal to its transpose. By default, simplifications occur before testing symmetry. They can be skipped using 'simplify=False'; while speeding things a bit, this may however induce false negatives. Example: >>> from sympy import Matrix >>> m = Matrix(2,2,[0, 1, 1, 2]) >>> m [0, 1] [1, 2] >>> m.is_symmetric() True >>> m = Matrix(2,2,[0, 1, 2, 0]) >>> m [0, 1] [2, 0] >>> m.is_symmetric() False >>> m = Matrix(2,3,[0, 0, 0, 0, 0, 0]) >>> m [0, 0, 0] [0, 0, 0] >>> m.is_symmetric() False >>> from sympy.abc import x, y >>> m = Matrix(3,3,[1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3]) >>> m [ 1, x**2 + 2*x + 1, y] [(x + 1)**2, 2, 0] [ y, 0, 3] >>> m.is_symmetric() True If the matrix is already simplified, you may speed-up is_symmetric() test by using 'simplify=False'. >>> m.is_symmetric(simplify=False) False >>> m1 = m.expand() >>> m1.is_symmetric(simplify=False) True """ if not self.is_square: return False if simplify: delta = self - self.transpose() delta.simplify() return delta == self.zeros((self.rows, self.cols)) else: return self == self.transpose()
[docs] def is_diagonal(self): """ Check if matrix is diagonal, that is matrix in which the entries outside the main diagonal are all zero. Example: >>> from sympy import Matrix, diag >>> m = Matrix(2,2,[1, 0, 0, 2]) >>> m [1, 0] [0, 2] >>> m.is_diagonal() True >>> m = Matrix(2,2,[1, 1, 0, 2]) >>> m [1, 1] [0, 2] >>> m.is_diagonal() False >>> m = diag(1, 2, 3) >>> m [1, 0, 0] [0, 2, 0] [0, 0, 3] >>> m.is_diagonal() True See also: .is_lower(), is_upper() .is_diagonalizable() """ for i in xrange(self.rows): for j in xrange(self.cols): if i != j and self[i, j] != 0: return False return True
def clone(self): return Matrix(self.rows, self.cols, lambda i, j: self[i, j])
[docs] def det(self, method="bareis"): """ Computes the matrix determinant using the method "method". Possible values for "method": bareis ... det_bareis berkowitz ... berkowitz_det """ if method == "bareis": return self.det_bareis() elif method == "berkowitz": return self.berkowitz_det() else: raise ValueError("Determinant method unrecognized")
[docs] def det_bareis(self): """Compute matrix determinant using Bareis' fraction-free algorithm which is an extension of the well known Gaussian elimination method. This approach is best suited for dense symbolic matrices and will result in a determinant with minimal number of fractions. It means that less term rewriting is needed on resulting formulae. TODO: Implement algorithm for sparse matrices (SFF). """ if not self.is_square: raise NonSquareMatrixError() M, n = self[:,:], self.rows if n == 1: det = M[0, 0] elif n == 2: det = M[0, 0]*M[1, 1] - M[0, 1]*M[1, 0] else: sign = 1 # track current sign in case of column swap for k in range(n-1): # look for a pivot in the current column # and assume det == 0 if none is found if M[k, k] == 0: for i in range(k+1, n): if M[i, k] != 0: M.row_swap(i, k) sign *= -1 break else: return S.Zero # proceed with Bareis' fraction-free (FF) # form of Gaussian elimination algorithm for i in range(k+1, n): for j in range(k+1, n): D = M[k, k]*M[i, j] - M[i, k]*M[k, j] if k > 0: D /= M[k-1, k-1] if D.is_Atom: M[i, j] = D else: M[i, j] = cancel(D) det = sign * M[n-1, n-1] return det.expand()
[docs] def adjugate(self, method="berkowitz"): """ Returns the adjugate matrix. Adjugate matrix is the transpose of the cofactor matrix. http://en.wikipedia.org/wiki/Adjugate See also: .cofactorMatrix(), .T """ return self.cofactorMatrix(method).T
[docs] def inverse_LU(self, iszerofunc=_iszero): """ Calculates the inverse using LU decomposition. """ return self.LUsolve(self.eye(self.rows), iszerofunc=_iszero)
[docs] def inverse_GE(self, iszerofunc=_iszero): """ Calculates the inverse using Gaussian elimination. """ if not self.is_square: raise NonSquareMatrixError() if self.det() == 0: raise ValueError("A Matrix must have non-zero determinant to invert.") big = self.row_join(self.eye(self.rows)) red = big.rref(iszerofunc=iszerofunc) return red[0][:,big.rows:]
[docs] def inverse_ADJ(self): """ Calculates the inverse using the adjugate matrix and a determinant. """ if not self.is_square: raise NonSquareMatrixError() d = self.berkowitz_det() if d == 0: raise ValueError("A Matrix must have non-zero determinant to invert.") return self.adjugate()/d
[docs] def rref(self,simplified=False, iszerofunc=_iszero, simplify=sympy_simplify): """ Take any matrix and return reduced row-echelon form and indices of pivot vars To simplify elements before finding nonzero pivots set simplified=True. To set a custom simplify function, use the simplify keyword argument. """ # TODO: rewrite inverse_GE to use this pivots, r = 0, self[:,:] # pivot: index of next row to contain a pivot pivotlist = [] # indices of pivot variables (non-free) for i in range(r.cols): if pivots == r.rows: break if simplified: r[pivots,i] = simplify(r[pivots,i]) if iszerofunc(r[pivots,i]): for k in range(pivots, r.rows): if simplified and k > pivots: r[k,i] = simplify(r[k,i]) if not iszerofunc(r[k,i]): break if k == r.rows - 1 and iszerofunc(r[k,i]): continue r.row_swap(pivots,k) scale = r[pivots,i] r.row(pivots, lambda x, _: x/scale) for j in range(r.rows): if j == pivots: continue scale = r[j,i] r.row(j, lambda x, k: x - scale*r[pivots,k]) pivotlist.append(i) pivots += 1 return r, pivotlist
[docs] def nullspace(self,simplified=False): """ Returns list of vectors (Matrix objects) that span nullspace of self """ reduced, pivots = self.rref(simplified) basis = [] # create a set of vectors for the basis for i in range(self.cols - len(pivots)): basis.append(zeros((self.cols, 1))) # contains the variable index to which the vector corresponds basiskey, cur = [-1]*len(basis), 0 for i in range(self.cols): if i not in pivots: basiskey[cur] = i cur += 1 for i in range(self.cols): if i not in pivots: # free var, just set vector's ith place to 1 basis[basiskey.index(i)][i,0] = 1 else: # add negative of nonpivot entry to corr vector for j in range(i+1, self.cols): line = pivots.index(i) if reduced[line, j] != 0: if j in pivots: # XXX: Is this the correct error? raise NotImplementedError("Could not compute the nullspace of `self`.") basis[basiskey.index(j)][i,0] = -1 * reduced[line, j] return basis
[docs] def berkowitz(self): """The Berkowitz algorithm. Given N x N matrix with symbolic content, compute efficiently coefficients of characteristic polynomials of 'self' and all its square sub-matrices composed by removing both i-th row and column, without division in the ground domain. This method is particularly useful for computing determinant, principal minors and characteristic polynomial, when 'self' has complicated coefficients e.g. polynomials. Semi-direct usage of this algorithm is also important in computing efficiently sub-resultant PRS. Assuming that M is a square matrix of dimension N x N and I is N x N identity matrix, then the following following definition of characteristic polynomial is begin used: charpoly(M) = det(t*I - M) As a consequence, all polynomials generated by Berkowitz algorithm are monic. >>> from sympy import Matrix >>> from sympy.abc import x, y, z >>> M = Matrix([ [x,y,z], [1,0,0], [y,z,x] ]) >>> p, q, r = M.berkowitz() >>> print p # 1 x 1 M's sub-matrix (1, -x) >>> print q # 2 x 2 M's sub-matrix (1, -x, -y) >>> print r # 3 x 3 M's sub-matrix (1, -2*x, x**2 - y*z - y, x*y - z**2) For more information on the implemented algorithm refer to: [1] S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, ACM, Information Processing Letters 18, 1984, pp. 147-150 [2] M. Keber, Division-Free computation of sub-resultants using Bezout matrices, Tech. Report MPI-I-2006-1-006, Saarbrucken, 2006 """ if not self.is_square: raise NonSquareMatrixError() A, N = self, self.rows transforms = [0] * (N-1) for n in xrange(N, 1, -1): T, k = zeros((n+1,n)), n - 1 R, C = -A[k,:k], A[:k,k] A, a = A[:k,:k], -A[k,k] items = [ C ] for i in xrange(0, n-2): items.append(A * items[i]) for i, B in enumerate(items): items[i] = (R * B)[0,0] items = [ S.One, a ] + items for i in xrange(n): T[i:,i] = items[:n-i+1] transforms[k-1] = T polys = [ Matrix([S.One, -A[0,0]]) ] for i, T in enumerate(transforms): polys.append(T * polys[i]) return tuple(map(tuple, polys))
[docs] def berkowitz_det(self): """Computes determinant using Berkowitz method.""" poly = self.berkowitz()[-1] sign = (-1)**(len(poly)-1) return sign * poly[-1]
[docs] def berkowitz_minors(self): """Computes principal minors using Berkowitz method.""" sign, minors = S.NegativeOne, [] for poly in self.berkowitz(): minors.append(sign*poly[-1]) sign = -sign return tuple(minors)
[docs] def berkowitz_charpoly(self, x, simplify=sympy_simplify): """Computes characteristic polynomial minors using Berkowitz method.""" return Poly(map(simplify, self.berkowitz()[-1]), x)
charpoly = berkowitz_charpoly
[docs] def berkowitz_eigenvals(self, **flags): """Computes eigenvalues of a Matrix using Berkowitz method. """ return roots(self.berkowitz_charpoly(Dummy('x')), **flags)
eigenvals = berkowitz_eigenvals
[docs] def eigenvects(self, **flags): """Return list of triples (eigenval, multiplicity, basis).""" if 'multiple' in flags: del flags['multiple'] out, vlist = [], self.eigenvals(**flags) for r, k in vlist.iteritems(): tmp = self - eye(self.rows)*r basis = tmp.nullspace() # whether tmp.is_symbolic() is True or False, it is possible that # the basis will come back as [] in which case simplification is # necessary. if not basis: # The nullspace routine failed, try it again with simplification basis = tmp.nullspace(simplified=True) if not basis: raise NotImplementedError("Can't evaluate eigenvector for eigenvalue %s" % r) out.append((r, k, basis)) return out
[docs] def singular_values(self): """ Compute the singular values of a Matrix >>> from sympy import Matrix, Symbol, eye >>> x = Symbol('x', real=True) >>> A = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]]) >>> print A.singular_values() [1, (x**2 + 1)**(1/2), 0] """ # Compute eigenvalues of A.H A valmultpairs = (self.H*self).eigenvals() # Expands result from eigenvals into a simple list vals = [] for k,v in valmultpairs.items(): vals += [sqrt(k)]*v # dangerous! same k in several spots! # If sorting makes sense then sort if all(val.is_number for val in vals): vals.sort(reverse=True) # sort them in descending order return vals
[docs] def condition_number(self): """ Returns the condition number of a matrix. This is the maximum singular value divided by the minimum singular value >>> from sympy import Matrix, S >>> A = Matrix([[1, 0, 0], [0, 10, 0], [0,0,S.One/10]]) >>> print A.condition_number() 100 """ singularvalues = self.singular_values() return Max(*singularvalues) / Min(*singularvalues)
[docs] def fill(self, value): """Fill the matrix with the scalar value.""" self.mat = [value]*len(self)
def __getattr__(self, attr): if attr in ('diff','integrate','limit'): def doit(*args): item_doit = lambda item: getattr(item, attr)(*args) return self.applyfunc( item_doit ) return doit else: raise AttributeError("Matrix has no attribute %s." % attr) def integrate(self, *args): return Matrix(self.rows, self.cols, lambda i, j: self[i, j].integrate(*args)) def limit(self, *args): return Matrix(self.rows, self.cols, lambda i, j: self[i, j].limit(*args)) def diff(self, *args): return Matrix(self.rows, self.cols, lambda i, j: self[i, j].diff(*args))
[docs] def vec(self): """ Return the Matrix converted into a one column matrix by stacking columns >>> from sympy import Matrix >>> m=Matrix([ [1,3], [2,4] ]) >>> m [1, 3] [2, 4] >>> m.vec() [1] [2] [3] [4] """ return Matrix(len(self), 1, self.transpose().mat)
[docs] def vech(self, diagonal=True, check_symmetry=True): """ Return the unique elements of a symmetric Matrix as a one column matrix by stacking the elements in the lower triangle. Arguments: diagonal -- include the diagonal cells of self or not check_symmetry -- checks symmetry of self but not completely reliably >>> from sympy import Matrix >>> m=Matrix([ [1,2], [2,3] ]) >>> m [1, 2] [2, 3] >>> m.vech() [1] [2] [3] >>> m.vech(diagonal=False) [2] """ c = self.cols if c != self.rows: raise ShapeError("Matrix must be square") if check_symmetry: self.simplify() if self != self.transpose(): raise ValueError("Matrix appears to be asymmetric; consider check_symmetry=False") count = 0 if diagonal: v = zeros( (c * (c + 1) // 2, 1) ) for j in xrange(c): for i in xrange(j,c): v[count] = self[i,j] count += 1 else: v = zeros( (c * (c - 1) // 2, 1) ) for j in xrange(c): for i in xrange(j+1,c): v[count] = self[i,j] count += 1 return v
[docs] def get_diag_blocks(self): """Obtains the square sub-matrices on the main diagonal of a square matrix. Useful for inverting symbolic matrices or solving systems of linear equations which may be decoupled by having a block diagonal structure. Example: >>> from sympy import Matrix, symbols >>> from sympy.abc import x, y, z >>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]]) >>> a1, a2, a3 = A.get_diag_blocks() >>> a1 [1, 3] [y, z**2] >>> a2 [x] >>> a3 [0] >>> """ sub_blocks = [] def recurse_sub_blocks(M): i = 1 while i <= M.shape[0]: if i == 1: to_the_right = M[0, i:] to_the_bottom = M[i:, 0] else: to_the_right = M[0:i, i:] to_the_bottom = M[i:, 0:i] if any(to_the_right) or any(to_the_bottom): i += 1 continue else: sub_blocks.append(M[0:i, 0:i]) if M.shape == M[0:i, 0:i].shape: return else: recurse_sub_blocks(M[i:, i:]) return recurse_sub_blocks(self) return sub_blocks
[docs] def diagonalize(self, reals_only = False): """ Return diagonalized matrix D and transformation P such as D = P^-1 * M * P where M is current matrix. Example: >>> from sympy import Matrix >>> m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2]) >>> m [1, 2, 0] [0, 3, 0] [2, -4, 2] >>> (P, D) = m.diagonalize() >>> D [1, 0, 0] [0, 2, 0] [0, 0, 3] >>> P [-1/2, 0, -1/2] [ 0, 0, -1/2] [ 1, 1, 1] >>> P.inv() * m * P [1, 0, 0] [0, 2, 0] [0, 0, 3] See also: .is_diagonalizable(), .is_diagonal() """ if not self.is_square: raise NonSquareMatrixError() if not self.is_diagonalizable(reals_only, False): self._diagonalize_clear_subproducts() raise MatrixError("Matrix is not diagonalizable") else: if self._eigenvects == None: self._eigenvects = self.eigenvects() diagvals = [] P = Matrix(self.rows, 0, []) for eigenval, multiplicity, vects in self._eigenvects: for k in range(multiplicity): diagvals.append(eigenval) vec = vects[k] P = P.col_insert(P.cols, vec) D = diag(*diagvals) self._diagonalize_clear_subproducts() return (P, D)
[docs] def is_diagonalizable(self, reals_only = False, clear_subproducts=True): """ Check if matrix is diagonalizable. If reals_only==True then check that diagonalized matrix consists of the only not complex values. Some subproducts could be used further in other methods to avoid double calculations, By default (if clear_subproducts==True) they will be deleted. Example: >>> from sympy import Matrix >>> m = Matrix(3,3,[1, 2, 0, 0, 3, 0, 2, -4, 2]) >>> m [1, 2, 0] [0, 3, 0] [2, -4, 2] >>> m.is_diagonalizable() True >>> m = Matrix(2,2,[0, 1, 0, 0]) >>> m [0, 1] [0, 0] >>> m.is_diagonalizable() False >>> m = Matrix(2,2,[0, 1, -1, 0]) >>> m [ 0, 1] [-1, 0] >>> m.is_diagonalizable() True >>> m.is_diagonalizable(True) False """ if not self.is_square: return False res = False self._is_symbolic = self.is_symbolic() self._is_symmetric = self.is_symmetric() self._eigenvects = None #if self._is_symbolic: # self._diagonalize_clear_subproducts() # raise NotImplementedError("Symbolic matrices are not implemented for diagonalization yet") self._eigenvects = self.eigenvects() all_iscorrect = True for eigenval, multiplicity, vects in self._eigenvects: if len(vects) != multiplicity: all_iscorrect = False break elif reals_only and not eigenval.is_real: all_iscorrect = False break res = all_iscorrect if clear_subproducts: self._diagonalize_clear_subproducts() return res
def _diagonalize_clear_subproducts(self): del self._is_symbolic del self._is_symmetric del self._eigenvects
[docs] def jordan_form(self, calc_transformation = True): """ Return Jordan form J of current matrix. If calc_transformation is specified as False, then transformation P such that J = P^-1 * M * P will not be calculated. Note: Calculation of transformation P is not implemented yet Example: >>> from sympy import Matrix >>> m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5]) >>> m [ 6, 5, -2, -3] [-3, -1, 3, 3] [ 2, 1, -2, -3] [-1, 1, 5, 5] >>> (P, J) = m.jordan_form() >>> J [2, 1, 0, 0] [0, 2, 0, 0] [0, 0, 2, 1] [0, 0, 0, 2] See also: jordan_cells() """ (P, Jcells) = self.jordan_cells(calc_transformation) J = diag(*Jcells) return (P, J)
[docs] def jordan_cells(self, calc_transformation = True): """ Return a list of Jordan cells of current matrix. This list shape Jordan matrix J. If calc_transformation is specified as False, then transformation P such that J = P^-1 * M * P will not be calculated. Note: Calculation of transformation P is not implemented yet Example: >>> from sympy import Matrix >>> m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5]) >>> m [ 6, 5, -2, -3] [-3, -1, 3, 3] [ 2, 1, -2, -3] [-1, 1, 5, 5] >>> (P, Jcells) = m.jordan_cells() >>> Jcells[0] [2, 1] [0, 2] >>> Jcells[1] [2, 1] [0, 2] See also: jordan_form() """ if not self.is_square: raise NonSquareMatrixError() _eigenvects = self.eigenvects() Jcells = [] for eigenval, multiplicity, vects in _eigenvects: geometrical = len(vects) if geometrical == multiplicity: Jcell = diag( *([eigenval] * multiplicity)) Jcells.append(Jcell) elif geometrical==0: raise MatrixError("Matrix has the eigen vector with geometrical multiplicity equal zero.") else: sizes = self._jordan_split(multiplicity, geometrical) cells = [] for size in sizes: cell = jordan_cell(eigenval, size) cells.append(cell) Jcells += cells return (None, Jcells)
def _jordan_split(self, algebraical, geometrical): "return a list which sum is equal to 'algebraical' and length is equal to 'geometrical'" n1 = algebraical // geometrical res = [n1] * geometrical res[len(res)-1] += algebraical % geometrical assert sum(res) == algebraical return res
[docs] def has(self, *patterns): """ Test whether any subexpression matches any of the patterns. Examples: >>> from sympy import Matrix, Float >>> from sympy.abc import x, y >>> A = Matrix(((1, x), (0.2, 3))) >>> A.has(x) True >>> A.has(y) False >>> A.has(Float) True """ return any(a.has(*patterns) for a in self.mat)
def matrix_multiply(A, B): """ Matrix product A*B. A and B must be of appropriate dimensions. If A is an m x k matrix, and B is a k x n matrix, the product will be an m x n matrix. Example: >>> from sympy import Matrix >>> A = Matrix([[1, 2, 3], [4, 5, 6]]) >>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> A*B [30, 36, 42] [66, 81, 96] >>> B*A Traceback (most recent call last): ... ShapeError >>> """ # The following implmentation is equivalent, but about 5% slower #ma, na = A.shape #mb, nb = B.shape # #if na != mb: # raise ShapeError() #product = Matrix(ma, nb, lambda i,j: 0) #for i in xrange(ma): # for j in xrange(nb): # s = 0 # for k in range(na): # s += A[i, k]*B[k, j] # product[i, j] = s #return product if A.shape[1] != B.shape[0]: raise ShapeError() blst = B.T.tolist() alst = A.tolist() return Matrix(A.shape[0], B.shape[1], lambda i, j: reduce(lambda k, l: k+l, map(lambda n, m: n*m, alst[i], blst[j]))) def matrix_multiply_elementwise(A, B): """Return the Hadamard product (elementwise product) of A and B >>> import sympy >>> A = sympy.Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = sympy.Matrix([[1, 10, 100], [100, 10, 1]]) >>> print sympy.matrices.matrix_multiply_elementwise(A, B) [ 0, 10, 200] [300, 40, 5] """ if A.shape != B.shape: raise ShapeError() shape = A.shape return Matrix(shape[0], shape[1], lambda i, j: A[i,j] * B[i, j]) def matrix_add(A,B): """Return A+B""" if A.shape != B.shape: raise ShapeError() alst = A.tolist() blst = B.tolist() ret = [0]*A.shape[0] for i in xrange(A.shape[0]): ret[i] = map(lambda j,k: j+k, alst[i], blst[i]) return Matrix(ret) def zeros(dims): """Create zero matrix of dimensions dims = (d1,d2)""" n, m = _dims_to_nm(dims) return Matrix(n, m, [S.Zero]*m*n) def ones(dims): """Create all-one matrix of dimensions dims = (d1,d2)""" n, m = _dims_to_nm( dims ) return Matrix(n, m, [S.One]*m*n) def eye(n): """Create square identity matrix n x n See also: diag() """ n = int(n) out = zeros(n) for i in range(n): out[i, i] = S.One return out def diag(*values): """Create diagonal matrix from a list as a diagonal values. Arguments might be matrices too, in case of it they are fitted in result matrix Example: >>> from sympy.matrices import diag, Matrix >>> diag(1, 2, 3) [1, 0, 0] [0, 2, 0] [0, 0, 3] >>> from sympy.abc import x, y, z >>> a = Matrix([x, y, z]) >>> b = Matrix([[1, 2], [3, 4]]) >>> c = Matrix([[5, 6]]) >>> diag(a, 7, b, c) [x, 0, 0, 0, 0, 0] [y, 0, 0, 0, 0, 0] [z, 0, 0, 0, 0, 0] [0, 7, 0, 0, 0, 0] [0, 0, 1, 2, 0, 0] [0, 0, 3, 4, 0, 0] [0, 0, 0, 0, 5, 6] See also: eye() """ rows = 0 cols = 0 for m in values: if isinstance(m, Matrix): rows += m.rows cols += m.cols else: rows += 1 cols += 1 res = zeros((rows, cols)) i_row = 0 i_col = 0 for m in values: if isinstance(m, Matrix): res[i_row:i_row + m.rows, i_col:i_col + m.cols] = m i_row += m.rows i_col += m.cols else: res[i_row, i_col] = m i_row += 1 i_col += 1 return res def block_diag(matrices): """ Warning: this function is deprecated. See .diag() """ import warnings warnings.warn("block_diag() is deprecated, use diag() instead", DeprecationWarning) return diag(*matrices) def jordan_cell(eigenval, n): """ Create matrix of Jordan cell kind: Example: >>> from sympy.matrices.matrices import jordan_cell >>> from sympy.abc import x >>> jordan_cell(x, 4) [x, 1, 0, 0] [0, x, 1, 0] [0, 0, x, 1] [0, 0, 0, x] """ n = int(n) out = zeros(n) for i in range(n-1): out[i, i] = eigenval out[i, i+1] = S.One out[n-1, n-1] = eigenval return out def randMatrix(r,c,min=0,max=99,seed=[]): """Create random matrix r x c""" if seed == []: prng = random.Random() # use system time else: prng = random.Random(seed) return Matrix(r,c,lambda i,j: prng.randint(min,max)) def hessian(f, varlist): """Compute Hessian matrix for a function f see: http://en.wikipedia.org/wiki/Hessian_matrix """ # f is the expression representing a function f, return regular matrix if is_sequence(varlist): m = len(varlist) if not m: raise ShapeError("`len(varlist)` must not be zero.") elif isinstance(varlist, Matrix): m = varlist.cols if not m: raise ShapeError("`varlist.cols` must not be zero.") if varlist.rows != 1: raise ShapeError("`varlist` must be a row vector.") else: raise ValueError("Improper variable list in hessian function") if not getattr(f, 'diff'): # check differentiability raise ValueError("Function `f` (%s) is not differentiable" % f) out = zeros(m) for i in range(m): for j in range(i,m): out[i,j] = f.diff(varlist[i]).diff(varlist[j]) for i in range(m): for j in range(i): out[i,j] = out[j,i] return out def GramSchmidt(vlist, orthog=False): out = [] m = len(vlist) for i in range(m): tmp = vlist[i] for j in range(i): tmp -= vlist[i].project(out[j]) if tmp == Matrix([[0,0,0]]): raise ValueError("GramSchmidt: vector set not linearly independent") out.append(tmp) if orthog: for i in range(len(out)): out[i] = out[i].normalized() return out def wronskian(functions, var, method='bareis'): """Compute Wronskian for [] of functions | f1 f2 ... fn | | f1' f2' ... fn' | | . . . . | W(f1,...,fn) = | . . . . | | . . . . | | (n) (n) (n) | | D (f1) D (f2) ... D (fn) | see: http://en.wikipedia.org/wiki/Wronskian """ for index in xrange(0, len(functions)): functions[index] = sympify(functions[index]) n = len(functions) if n == 0: return 1 W = Matrix(n, n, lambda i,j: functions[i].diff(var, j) ) return W.det(method) def casoratian(seqs, n, zero=True): """Given linear difference operator L of order 'k' and homogeneous equation Ly = 0 we want to compute kernel of L, which is a set of 'k' sequences: a(n), b(n), ... z(n). Solutions of L are linearly independent iff their Casoratian, denoted as C(a, b, ..., z), do not vanish for n = 0. Casoratian is defined by k x k determinant: + a(n) b(n) . . . z(n) + | a(n+1) b(n+1) . . . z(n+1) | | . . . . | | . . . . | | . . . . | + a(n+k-1) b(n+k-1) . . . z(n+k-1) + It proves very useful in rsolve_hyper() where it is applied to a generating set of a recurrence to factor out linearly dependent solutions and return a basis. >>> from sympy import Symbol, casoratian, factorial >>> n = Symbol('n', integer=True) Exponential and factorial are linearly independent: >>> casoratian([2**n, factorial(n)], n) != 0 True """ seqs = map(sympify, seqs) if not zero: f = lambda i, j: seqs[j].subs(n, n+i) else: f = lambda i, j: seqs[j].subs(n, i) k = len(seqs) return Matrix(k, k, f).det() # Add sympify converters def _matrix_sympify(matrix): raise SympifyError('Matrix cannot be sympified') converter[Matrix] = _matrix_sympify del _matrix_sympify class SparseMatrix(Matrix): """Sparse matrix""" def __init__(self, *args): if len(args) == 3 and callable(args[2]): op = args[2] if not isinstance(args[0], (int, Integer)) or not isinstance(args[1], (int, Integer)): raise TypeError("`args[0]` and `args[1]` must both be integers.") self.rows = args[0] self.cols = args[1] self.mat = {} for i in range(self.rows): for j in range(self.cols): value = sympify(op(i,j)) if value != 0: self.mat[(i,j)] = value elif len(args)==3 and isinstance(args[0],int) and \ isinstance(args[1],int) and is_sequence(args[2]): self.rows = args[0] self.cols = args[1] mat = args[2] self.mat = {} for i in range(self.rows): for j in range(self.cols): value = sympify(mat[i*self.cols+j]) if value != 0: self.mat[(i,j)] = value elif len(args)==3 and isinstance(args[0],int) and \ isinstance(args[1],int) and isinstance(args[2], dict): self.rows = args[0] self.cols = args[1] self.mat = {} # manual copy, copy.deepcopy() doesn't work for key in args[2].keys(): self.mat[key] = args[2][key] else: if len(args) == 1: mat = args[0] else: mat = args if not is_sequence(mat[0]): mat = [ [element] for element in mat ] self.rows = len(mat) self.cols = len(mat[0]) self.mat = {} for i in range(self.rows): if len(mat[i]) != self.cols: raise ValueError("All arguments must have the same length.") for j in range(self.cols): value = sympify(mat[i][j]) if value != 0: self.mat[(i,j)] = value def __getitem__(self, key): if isinstance(key, slice) or isinstance(key, int): lo, hi = self.slice2bounds(key, len(self)) L = [] for i in range(lo, hi): m,n = self.rowdecomp(i) if (m,n) in self.mat: L.append(self.mat[(m,n)]) else: L.append(0) if len(L) == 1: return L[0] else: return L if len(key) != 2: raise ValueError("`key` must be of length 2.") if isinstance(key[0], int) and isinstance(key[1], int): i,j=self.key2ij(key) if (i, j) in self.mat: return self.mat[(i,j)] else: return 0 elif isinstance(key[0], slice) or isinstance(key[1], slice): return self.submatrix(key) else: raise IndexError("Index out of range: a[%s]"%repr(key)) def rowdecomp(self, num): nmax = len(self) if not (0 <= num < nmax) or not (0 <= -num < nmax): raise ValueError("`num` must satisfy 0 <= `num` < `self.rows*" + "*self.cols` (%d) and 0 <= -num < " % nmax + "`self.rows*self.cols` (%d) to apply redecomp()." % nmax) i, j = 0, num while j >= self.cols: j -= self.cols i += 1 return i,j def __setitem__(self, key, value): # almost identical, need to test for 0 if len(key) != 2: raise ValueError("`key` must be of length 2.") if isinstance(key[0], slice) or isinstance(key[1], slice): if isinstance(value, Matrix): self.copyin_matrix(key, value) if is_sequence(value): self.copyin_list(key, value) else: i,j=self.key2ij(key) testval = sympify(value) if testval != 0: self.mat[(i,j)] = testval elif (i,j) in self.mat: del self.mat[(i,j)] def row_del(self, k): newD = {} for (i,j) in self.mat.keys(): if i==k: pass elif i > k: newD[i-1,j] = self.mat[i,j] else: newD[i,j] = self.mat[i,j] self.mat = newD self.rows -= 1 def col_del(self, k): newD = {} for (i,j) in self.mat.keys(): if j==k: pass elif j > k: newD[i,j-1] = self.mat[i,j] else: newD[i,j] = self.mat[i,j] self.mat = newD self.cols -= 1 def toMatrix(self): l = [] for i in range(self.rows): c = [] l.append(c) for j in range(self.cols): if (i, j) in self.mat: c.append(self[i, j]) else: c.append(0) return Matrix(l) def row_list(self): """ Returns a Row-sorted list of non-zero elements of the matrix. >>> from sympy.matrices import SparseMatrix >>> a=SparseMatrix((1,2),(3,4)) >>> a [1, 2] [3, 4] >>> a.RL [(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)] """ new=[] for i in range(self.rows): for j in range(self.cols): value = self[(i,j)] if value!=0: new.append((i,j,value)) return new RL = property(row_list,None,None,"Alternate faster representation") def col_list(self): """ Returns a Column-sorted list of non-zero elements of the matrix. >>> from sympy.matrices import SparseMatrix >>> a=SparseMatrix((1,2),(3,4)) >>> a [1, 2] [3, 4] >>> a.CL [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)] """ new=[] for j in range(self.cols): for i in range(self.rows): value = self[(i,j)] if value!=0: new.append((i,j,value)) return new CL = property(col_list,None,None,"Alternate faster representation") def transpose(self): """ Returns the transposed SparseMatrix of this SparseMatrix >>> from sympy.matrices import SparseMatrix >>> a = SparseMatrix((1,2),(3,4)) >>> a [1, 2] [3, 4] >>> a.T [1, 3] [2, 4] """ tran = SparseMatrix(self.cols,self.rows,{}) for key,value in self.mat.iteritems(): tran.mat[key[1],key[0]]=value return tran T = property(transpose,None,None,"Matrix transposition.") def __add__(self, other): if isinstance(other, SparseMatrix): return self.add(other) else: raise NotImplementedError("Only SparseMatrix + SparseMatrix supported") def __radd__(self, other): if isinstance(other, SparseMatrix): return self.add(other) else: raise NotImplementedError("Only SparseMatrix + SparseMatrix supported") def add(self, other): """ Add two sparse matrices with dictionary representation. >>> from sympy.matrices.matrices import SparseMatrix >>> A = SparseMatrix(5, 5, lambda i, j : i * j + i) >>> A [0, 0, 0, 0, 0] [1, 2, 3, 4, 5] [2, 4, 6, 8, 10] [3, 6, 9, 12, 15] [4, 8, 12, 16, 20] >>> B = SparseMatrix(5, 5, lambda i, j : i + 2 * j) >>> B [0, 2, 4, 6, 8] [1, 3, 5, 7, 9] [2, 4, 6, 8, 10] [3, 5, 7, 9, 11] [4, 6, 8, 10, 12] >>> A + B [0, 2, 4, 6, 8] [2, 5, 8, 11, 14] [4, 8, 12, 16, 20] [6, 11, 16, 21, 26] [8, 14, 20, 26, 32] """ if self.shape != other.shape: raise ShapeError() a, b = self.mat.keys(), other.mat.keys() a.sort() b.sort() i = j = 0 c = {} while i < len(a) or j < len(b): if j >= len(b) or (i < len(a) and a[i] < b[j]): c[a[i]] = self.mat[a[i]] i = i + 1 continue elif i >= len(a) or (j < len(b) and a[i] > b[j]): c[b[j]] = other.mat[b[j]] j = j + 1 continue else: c[a[i]] = self.mat[a[i]] + other.mat[b[j]] i = i + 1 j = j + 1 return SparseMatrix(self.rows, self.cols, c) # from here to end all functions are same as in matrices.py # with Matrix replaced with SparseMatrix def copyin_list(self, key, value): if not is_sequence(value): raise TypeError("`value` must be of type list or tuple.") self.copyin_matrix(key, SparseMatrix(value)) def multiply(self,b): """Returns self*b """ def dotprod(a,b,i,j): if a.cols != b.rows: raise ShapeError("`self.cols` must equal `b.rows`.") r=0 for x in range(a.cols): r+=a[i,x]*b[x,j] return r r = SparseMatrix(self.rows, b.cols, lambda i,j: dotprod(self,b,i,j)) if r.rows == 1 and r.cols ==1: return r[0,0] return r def submatrix(self, keys): if not isinstance(keys[0], slice) and not isinstance(keys[1], slice): raise TypeError("Both elements of `keys` must be slice objects.") rlo, rhi = self.slice2bounds(keys[0], self.rows) clo, chi = self.slice2bounds(keys[1], self.cols) if not ( 0<=rlo<=rhi and 0<=clo<=chi ): raise IndexError("Slice indices out of range: a[%s]"%repr(keys)) return SparseMatrix(rhi-rlo, chi-clo, lambda i,j: self[i+rlo, j+clo]) def reshape(self, _rows, _cols): if len(self) != _rows*_cols: print "Invalid reshape parameters %d %d" % (_rows, _cols) newD = {} for i in range(_rows): for j in range(_cols): m,n = self.rowdecomp(i*_cols + j) if (m,n) in self.mat: newD[(i,j)] = self.mat[(m,n)] return SparseMatrix(_rows, _cols, newD) def cross(self, b): if not is_sequence(b, include=Matrix): raise TypeError("`b` must be an ordered iterable or Matrix, not %s." % type(b)) if not (self.rows == 1 and self.cols == 3 or \ self.rows == 3 and self.cols == 1 ) and \ (b.rows == 1 and b.cols == 3 or \ b.rows == 3 and b.cols == 1): raise ShapeError("Dimensions incorrect for cross product") else: return SparseMatrix(1,3,((self[1]*b[2] - self[2]*b[1]), (self[2]*b[0] - self[0]*b[2]), (self[0]*b[1] - self[1]*b[0]))) def zeros(self, dims): """Returns a dims = (d1,d2) matrix of zeros.""" n, m = _dims_to_nm( dims ) return SparseMatrix(n,m,{}) def eye(self, n): tmp = SparseMatrix(n,n,lambda i,j:0) for i in range(tmp.rows): tmp[i,i] = 1 return tmp def list2numpy(l): """Converts python list of SymPy expressions to a NumPy array.""" from numpy import empty a = empty(len(l), dtype=object) for i, s in enumerate(l): a[i] = s return a def matrix2numpy(m): """Converts SymPy's matrix to a NumPy array.""" from numpy import empty a = empty(m.shape, dtype=object) for i in range(m.rows): for j in range(m.cols): a[i, j] = m[i, j] return a def a2idx(a): """ Tries to convert "a" to an index, returns None on failure. The result of a2idx() (if not None) can be safely used as an index to arrays/matrices. """ if hasattr(a, "__int__"): return int(a) if hasattr(a, "__index__"): return a.__index__() def symarray(prefix, shape): """Create a numpy ndarray of symbols (as an object array). The created symbols are named prefix_i1_i2_... You should thus provide a non-empty prefix if you want your symbols to be unique for different output arrays, as Sympy symbols with identical names are the same object. Parameters ---------- prefix : string A prefix prepended to the name of every symbol. shape : int or tuple Shape of the created array. If an int, the array is one-dimensional; for more than one dimension the shape must be a tuple. Examples -------- >> from sympy import symarray >> symarray('', 3) [_0 _1 _2] If you want multiple symarrays to contain distinct symbols, you *must* provide unique prefixes: >> a = symarray('', 3) >> b = symarray('', 3) >> a[0] is b[0] True >> a = symarray('a', 3) >> b = symarray('b', 3) >> a[0] is b[0] False Creating symarrays with a prefix: >> symarray('a', 3) [a_0 a_1 a_2] For more than one dimension, the shape must be given as a tuple: >> symarray('a', (2,3)) [[a_0_0 a_0_1 a_0_2] [a_1_0 a_1_1 a_1_2]] >> symarray('a', (2,3,2)) [[[a_0_0_0 a_0_0_1] [a_0_1_0 a_0_1_1] [a_0_2_0 a_0_2_1]] <BLANKLINE> [[a_1_0_0 a_1_0_1] [a_1_1_0 a_1_1_1] [a_1_2_0 a_1_2_1]]] """ try: import numpy as np except ImportError: raise ImportError("symarray requires numpy to be installed") arr = np.empty(shape, dtype=object) for index in np.ndindex(shape): arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index)))) return arr def _separate_eig_results(res): eigvals = [item[0] for item in res] multiplicities = [item[1] for item in res] eigvals = flatten([[val]*mult for val, mult in zip(eigVals, multiplicities)]) eigvects = flatten([item[2] for item in res]) return eigvals, eigvects