ó ¡¼™\c@sdZddlZddlmZddlmZddlmZddlm Z m Z m Z m Z m Z mZddlmZedd d d d d ƒjƒd„Zd„Zdd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„ZdS(s^ Solution of equations using dense matrices. The dense matrix is stored as a list of lists. iÿÿÿÿN(trange(tisqrt(tsymbols(taugmenttcoltconjugate_transposeteyetrowaddtrowmul(tSymPyDeprecationWarningtfeaturet densesolvetissuei—1tdeprecated_since_versions1.1cCsÔtj|ƒ}t|ƒ}x²t|ƒD]¤}|||dkrx|||dkrxt||d||||ƒnxQt|d|ƒD]<}|||dkrŒt|||||| |ƒqŒqŒWq(W|S(s¦ 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 ii(tcopytdeepcopytlenRRR(tmatlisttKtresult_matlisttnrowtitj((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyt row_echelons ("*cCs’tj|ƒ}t||ƒ}t|ƒ}xat|ƒD]S}|||dkr7x6t|ƒD]%}t|||||| |ƒq^Wq7q7W|S(s{ 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 i(RRRRRR(RRRRRR((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytrref8s *ic CsÎt|ƒ}t||ƒtj|ƒ}}x–t|ƒD]ˆ}xt|d|ƒD]j}|||dkrR|||||||||>> 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 ii(RRRRRR(RRtreverseRt new_matlist1t new_matlist2RR((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytLUWs "6c Cstj|ƒ}t|ƒ}t||ƒ}xÂt|ƒD]´}x«t|dƒD]™}|j}x2t|ƒD]$}|||||||7}qjW||krÁt||||ƒ|||>> 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 i(RRRRRtzeroRR( RRt new_matlistRtLRRtatk((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytcholeskyxs  " #.c Cstj|ƒ}t|ƒ}t||ƒt||ƒ}}xÈt|ƒD]º}x±t|dƒD]Ÿ}|j}x>t|ƒD]0} |||| ||| || | 7}qzW||kr×|||||||>> 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]]) i(RRRRRRR( RRRRRtDRRR R!((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytLDL™s  . .cCs(tj|ƒ}t||ƒ\}}|S(sÐ 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 (RRR(RRt copy_matlisttlower_triangletupper_triangle((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyR'»scCs.tj|ƒ}t||ddƒ\}}|S(sË 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 Ri(RRR(RRR%R&R'((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyR&ÖscCs=tj|ƒ}t|||ƒ}t||ƒ}t|dƒS(sø 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 iÿÿÿÿ(RRRRR(RtvariabletconstantRRt augmentedtsolution((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyt rref_solveñs!c Cs€tj|ƒ}t|ƒ}t||ƒ\}}gtd|ƒD]}|g^qA} t|| ||ƒt||| |ƒ|S(sÿ 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 sy:%i(RRRRRtforward_substitutiontbackward_substitution( RR(R)RRRRtURty((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytLU_solves" &c Cs€tj|ƒ}t|ƒ}t||ƒ\}}gtd|ƒD]}|g^qA} t|| ||ƒt||| |ƒ|S(s} 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 sy:%i(RRRR"RR-R.( RR(R)RRRRtLstarRR0((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pytcholesky_solveCs" &c Cs tj|ƒ}t|ƒ}g}xxt|ƒD]j}|j}x2t|ƒD]$} |||| || d7}qJW||d||||||d>> 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 i(RRRRR( R&R(R)Rtcopy_lower_triangleRtresultRR R((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyR-ns!  "*c Cs³tj|ƒ}t|ƒ}g}x‹tt|ƒƒD]w}|j}x?tt|d|ƒƒD]$} |||| || d7}q]W||d||||||d>> from sympy.matrices.densesolve import backward_substitution >>> from sympy import QQ >>> from sympy import Dummy >>> x, y, z = Dummy('x'), Dummy('y'), Dummy('z') >>> a = [ ... [QQ(2), QQ(-1), QQ(-2)], ... [QQ(0), QQ(4), QQ(-1)], ... [QQ(0), QQ(0), QQ(3)]] >>> variables = [ ... [x], ... [y], ... [z]] >>> constants = [ ... [QQ(-1)], ... [QQ(11)], ... [QQ(3)]] >>> backward_substitution(a, variables, constants, QQ) [[2], [3], [1]] See Also ======== LU_solve cholesky_solve ii(RRRtreversedRR( R'R(R)Rtcopy_upper_triangleRR5RR R((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyR.šs!   "*(t__doc__Rtsympy.core.compatibilityRtsympy.core.powerRtsympy.core.symbolRtsympy.matrices.densetoolsRRRRRRtsympy.utilities.exceptionsR twarnRRRR"R$R'R&R,R1R3R-R.(((s8lib/python2.7/site-packages/sympy/matrices/densesolve.pyts, .  !  ! ! "   ' + + ,