o 8Va-@sdZddlZddlmZddlmZddlmZmZm Z m Z m Z m Z ddl mZeddd d d d Zd dZd#ddZddZddZddZddZddZddZddZdd Zd!d"ZdS)$z^ Solution of equations using dense matrices. The dense matrix is stored as a list of lists. N)isqrt)symbols)augmentcolconjugate_transposeeyerowaddrowmul)SymPyDeprecationWarningZ densesolvei1z1.1)ZfeatureZissueZdeprecated_since_versioncCst|}t|}t|D]>}|||dkr,|||dkr,t||d||||t|d|D]}|||dkrJt|||||| |q3q |S)a Returns the row echelon form of a matrix with diagonal elements reduced to 1. Examples ======== >>> from sympy.matrices.densesolve import row_echelon >>> from sympy import QQ >>> a = [ ... [QQ(3), QQ(7), QQ(4)], ... [QQ(2), QQ(4), QQ(5)], ... [QQ(6), QQ(2), QQ(3)]] >>> row_echelon(a, QQ) [[1, 7/3, 4/3], [0, 1, -7/2], [0, 0, 1]] See Also ======== rref r)copydeepcopylenranger rmatlistKZresult_matlistnrowijr;/usr/lib/python3/dist-packages/sympy/matrices/densesolve.py row_echelons   rcCsft|}t||}t|}t|D]}|||dkr0t|D]}t|||||| |q q|S)a{ Returns the reduced row echelon form of a Matrix. Examples ======== >>> from sympy.matrices.densesolve import rref >>> from sympy import QQ >>> a = [ ... [QQ(1), QQ(2), QQ(1)], ... [QQ(-2), QQ(-3), QQ(1)], ... [QQ(3), QQ(5), QQ(0)]] >>> rref(a, QQ) [[1, 0, -5], [0, 1, 3], [0, 0, 0]] See Also ======== row_echelon r )r r rrrrrrrrrref7s    rc Cst|}t||t|}}t|D]7}t|d|D]-}|||dkrI|||||||||<t|||||| ||||qq||fS)a It computes the LU decomposition of a matrix and returns L and U matrices. Examples ======== >>> from sympy.matrices.densesolve import LU >>> from sympy import QQ >>> a = [ ... [QQ(1), QQ(2), QQ(3)], ... [QQ(2), QQ(-4), QQ(6)], ... [QQ(3), QQ(-9), QQ(-3)]] >>> LU(a, QQ) ([[1, 0, 0], [2, 1, 0], [3, 15/8, 1]], [[1, 2, 3], [0, -8, 0], [0, 0, -12]]) See Also ======== upper_triangle lower_triangle r r)rrr r rr)rrreverserZ new_matlist1Z new_matlist2rrrrrLUVs  &rc Cst|}t|}t||}t|D]H}t|dD]?}|j}t|D]}|||||||7}q#||krGt|||||||<q||||||||||<qq|t||fS)a Performs the cholesky decomposition of a Hermitian matrix and returns L and it's conjugate transpose. Examples ======== >>> from sympy.matrices.densesolve import cholesky >>> from sympy import QQ >>> cholesky([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], QQ) ([[5, 0, 0], [3, 3, 0], [-1, 1, 3]], [[5, 3, -1], [0, 3, 1], [0, 0, 3]]) See Also ======== cholesky_solve r )r r rrrzerorr) rr new_matlistrLrrakrrrcholeskyws    &r!c Cst|}t|}t||t||}}t|D]L}t|dD]C}|j}t|D]} |||| ||| || | 7}q)||krQ|||||||<q ||||||||||<q q||t||fS)a Performs the LDL decomposition of a hermitian matrix and returns L, D and transpose of L. Only applicable to rational entries. Examples ======== >>> from sympy.matrices.densesolve import LDL >>> from sympy import QQ >>> a = [ ... [QQ(4), QQ(12), QQ(-16)], ... [QQ(12), QQ(37), QQ(-43)], ... [QQ(-16), QQ(-43), QQ(98)]] >>> LDL(a, QQ) ([[1, 0, 0], [3, 1, 0], [-4, 5, 1]], [[4, 0, 0], [0, 1, 0], [0, 0, 9]], [[1, 3, -4], [0, 1, 5], [0, 0, 1]]) r )r r rrrrr) rrrrrDrrrr rrrLDLs   *&r#cCst|}t||\}}|S)a Transforms a given matrix to an upper triangle matrix by performing row operations on it. Examples ======== >>> from sympy.matrices.densesolve import upper_triangle >>> from sympy import QQ >>> a = [ ... [QQ(4,1), QQ(12,1), QQ(-16,1)], ... [QQ(12,1), QQ(37,1), QQ(-43,1)], ... [QQ(-16,1), QQ(-43,1), QQ(98,1)]] >>> upper_triangle(a, QQ) [[4, 12, -16], [0, 1, 5], [0, 0, 9]] See Also ======== LU r r rrrZ copy_matlistlower_triangleupper_trianglerrrr's r'cCs t|}t||dd\}}|S)a Transforms a given matrix to a lower triangle matrix by performing row operations on it. Examples ======== >>> from sympy.matrices.densesolve import lower_triangle >>> from sympy import QQ >>> a = [ ... [QQ(4,1), QQ(12,1), QQ(-16)], ... [QQ(12,1), QQ(37,1), QQ(-43,1)], ... [QQ(-16,1), QQ(-43,1), QQ(98,1)]] >>> lower_triangle(a, QQ) [[1, 0, 0], [3, 1, 0], [-4, 5, 1]] See Also ======== LU r )rr$r%rrrr&s r&cCs*t|}t|||}t||}t|dS)a Solves a system of equations using reduced row echelon form given a matrix of coefficients, a vector of variables and a vector of constants. Examples ======== >>> from sympy.matrices.densesolve import rref_solve >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> coefficients = [ ... [QQ(25), QQ(15), QQ(-5)], ... [QQ(15), QQ(18), QQ(0)], ... [QQ(-5), QQ(0), QQ(11)]] >>> constants = [ ... [QQ(2)], ... [QQ(3)], ... [QQ(1)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> rref_solve(coefficients, variables, constants, QQ) [[-1/225], [23/135], [4/45]] See Also ======== row_echelon augment )r r rrr)rvariableconstantrrZ augmentedZsolutionrrr rref_solves !   r+c CVt|}t|}t||\}}ddtd|D}t||||t|||||S)a Solves a system of equations using LU decomposition given a matrix of coefficients, a vector of variables and a vector of constants. Examples ======== >>> from sympy.matrices.densesolve import LU_solve >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> coefficients = [ ... [QQ(2), QQ(-1), QQ(-2)], ... [QQ(-4), QQ(6), QQ(3)], ... [QQ(-4), QQ(-2), QQ(8)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> constants = [ ... [QQ(-1)], ... [QQ(13)], ... [QQ(-6)]] >>> LU_solve(coefficients, variables, constants, QQ) [[2], [3], [1]] See Also ======== LU forward_substitution backward_substitution cSg|]}|gqSrr.0rrrr <zLU_solve..y:%i)r r rrrforward_substitutionbackward_substitution) rr)r*rrrrUyrrrLU_solve "r7c Cr,)a} Solves a system of equations using Cholesky decomposition given a matrix of coefficients, a vector of variables and a vector of constants. Examples ======== >>> from sympy.matrices.densesolve import cholesky_solve >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> coefficients = [ ... [QQ(25), QQ(15), QQ(-5)], ... [QQ(15), QQ(18), QQ(0)], ... [QQ(-5), QQ(0), QQ(11)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> coefficients = [ ... [QQ(2)], ... [QQ(3)], ... [QQ(1)]] >>> cholesky_solve([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], [[x], [y], [z]], [[QQ(2)], [QQ(3)], [QQ(1)]], QQ) [[-1/225], [23/135], [4/45]] See Also ======== cholesky forward_substitution backward_substitution cSr-rrr.rrrr0gr1z"cholesky_solve..r2)r r rr!rr3r4) rr)r*rrrrZLstarr6rrrcholesky_solveBr8r9c Csxt|}t|}t|D],}|j}t|D]}||||||d7}q||d||||||d<q |S)a Performs forward substitution given a lower triangular matrix, a vector of variables and a vector of constants. Examples ======== >>> from sympy.matrices.densesolve import forward_substitution >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> a = [ ... [QQ(1), QQ(0), QQ(0)], ... [QQ(-2), QQ(1), QQ(0)], ... [QQ(-2), QQ(-1), QQ(1)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> constants = [ ... [QQ(-1)], ... [QQ(13)], ... [QQ(-6)]] >>> forward_substitution(a, variables, constants, QQ) [[-1], [11], [3]] See Also ======== LU_solve cholesky_solve r)r r rrr) r&r)r*rZcopy_lower_trianglerrrrrrrr3ms !  &r3c Cst|}t|}tt|D]1}|j}tt|d|D]}||||||d7}q||d||||||d<q|S)a Performs forward substitution given a lower triangular matrix, a vector of variables and a vector constants. Examples ======== >>> from sympy.matrices.densesolve import backward_substitution >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> a = [ ... [QQ(2), QQ(-1), QQ(-2)], ... [QQ(0), QQ(4), QQ(-1)], ... [QQ(0), QQ(0), QQ(3)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> constants = [ ... [QQ(-1)], ... [QQ(11)], ... [QQ(3)]] >>> backward_substitution(a, variables, constants, QQ) [[2], [3], [1]] See Also ======== LU_solve cholesky_solve r r)r r rreversedrr) r'r)r*rZcopy_upper_trianglerrrrrrrr4s !&r4)r)__doc__r Zsympy.core.powerrZsympy.core.symbolrZsympy.matrices.densetoolsrrrrrr Zsympy.utilities.exceptionsr warnrrrr!r#r'r&r+r7r9r3r4rrrrs0    ! !!"'++ +