def LDL(matlist, K): """ 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]]) """ new_matlist = copy.deepcopy(matlist) nrow = len(new_matlist) L, D = eye(nrow, K), eye(nrow, K) for i in range(nrow): for j in range(i + 1): a = K.zero for k in range(j): a += L[i][k]*L[j][k]*D[k][k] if i == j: D[j][j] = new_matlist[j][j] - a else: L[i][j] = (new_matlist[i][j] - a)/D[j][j] return L, D, conjugate_transpose(L, K)
def LDL(matlist, K): """ 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]]) """ new_matlist = copy.deepcopy(matlist) nrow = len(new_matlist) L, D = eye(nrow, K), eye(nrow, K) for i in range(nrow): for j in range(i + 1): a = K.zero for k in range(j): a += L[i][k] * L[j][k] * D[k][k] if i == j: D[j][j] = new_matlist[j][j] - a else: L[i][j] = (new_matlist[i][j] - a) / D[j][j] return L, D, conjugate_transpose(L, K)
def LU(matlist, K, reverse=0): """ 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 """ nrow = len(matlist) new_matlist1, new_matlist2 = eye(nrow, K), copy.deepcopy(matlist) for i in range(nrow): for j in range(i + 1, nrow): if new_matlist2[j][i] != 0: new_matlist1[j][i] = new_matlist2[j][i] / new_matlist2[i][i] rowadd(new_matlist2, j, i, -new_matlist2[j][i] / new_matlist2[i][i], K) return new_matlist1, new_matlist2
def cholesky(matlist, K): """ 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 """ new_matlist = copy.deepcopy(matlist) nrow = len(new_matlist) L = eye(nrow, K) for i in range(nrow): for j in range(i + 1): a = K.zero for k in range(j): a += L[i][k] * L[j][k] if i == j: L[i][j] = isqrt(new_matlist[i][j] - a) else: L[i][j] = (new_matlist[i][j] - a) / L[j][j] return L, conjugate_transpose(L, K)
def LU(matlist, K, reverse = 0): """ 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 """ nrow = len(matlist) new_matlist1, new_matlist2 = eye(nrow, K), copy.deepcopy(matlist) for i in range(nrow): for j in range(i + 1, nrow): if (new_matlist2[j][i] != 0): new_matlist1[j][i] = new_matlist2[j][i]/new_matlist2[i][i] rowadd(new_matlist2, j, i, -new_matlist2[j][i]/new_matlist2[i][i], K) return new_matlist1, new_matlist2
def cholesky(matlist, K): """ 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 """ new_matlist = copy.deepcopy(matlist) nrow = len(new_matlist) L = eye(nrow, K) for i in range(nrow): for j in range(i + 1): a = K.zero for k in range(j): a += L[i][k]*L[j][k] if i == j: L[i][j] = int(sqrt(new_matlist[i][j] - a)) else: L[i][j] = (new_matlist[i][j] - a)/L[j][j] return L, conjugate_transpose(L, K)
def test_mulmatmat(): a = [[ZZ(3), ZZ(4)], [ZZ(5), ZZ(6)]] b = [[ZZ(1), ZZ(2)], [ZZ(7), ZZ(8)]] c = eye(2, ZZ) d = [[ZZ(6)], [ZZ(7)]] assert mulmatmat(a, b, ZZ) == [[ZZ(31), ZZ(38)], [ZZ(47), ZZ(58)]] assert mulmatmat(b, d, ZZ) == [[ZZ(20)], [ZZ(98)]]
def test_mulmatmat(): a = [[ZZ(3), ZZ(4)], [ZZ(5), ZZ(6)]] b = [[ZZ(1), ZZ(2)], [ZZ(7), ZZ(8)]] c = eye(2, ZZ) d = [[ZZ(6)], [ZZ(7)]] assert mulmatmat(a, b, ZZ) == [[ZZ(31), ZZ(38)], [ZZ(47), ZZ(58)]] assert mulmatmat(b, d, ZZ) == [[ZZ(20)], [ZZ(98)]]
def test_transpose(): a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] b = eye(4, ZZ) assert transpose(a, ZZ) == ([[ZZ(3), ZZ(2), ZZ(6)], [ZZ(7), ZZ(4), ZZ(2)], [ZZ(4), ZZ(5), ZZ(3)]]) assert transpose(b, ZZ) == b
def test_mulmatscaler(): a = eye(3, ZZ) b = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] assert mulmatscaler(a, ZZ(4), ZZ) == [[ZZ(4), ZZ(0), ZZ(0)], [ZZ(0), ZZ(4), ZZ(0)], [ZZ(0), ZZ(0), ZZ(4)]] assert mulmatscaler(b, ZZ(1), ZZ) == [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]]
def test_trace(): a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] b = eye(2, ZZ) assert trace(a, ZZ) == ZZ(10) assert trace(b, ZZ) == ZZ(2)
def test_transpose(): a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] b = eye(4, ZZ) assert transpose(a, ZZ) == ([[ZZ(3), ZZ(2), ZZ(6)], [ZZ(7), ZZ(4), ZZ(2)], [ZZ(4), ZZ(5), ZZ(3)]]) assert transpose(b, ZZ) == b
def test_trace(): a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] b = eye(2, ZZ) assert trace(a, ZZ) == ZZ(10) assert trace(b, ZZ) == ZZ(2)
def test_mulmatscaler(): a = eye(3, ZZ) b = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]] assert mulmatscaler(a, ZZ(4), ZZ) == [[ZZ(4), ZZ(0), ZZ(0)], [ZZ(0), ZZ(4), ZZ(0)], [ZZ(0), ZZ(0), ZZ(4)]] assert mulmatscaler(b, ZZ(1), ZZ) == [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]]