Many functions of Julia for handling vectors and matrices are similar to those of MATLAB. Update B as alpha*A*B or one of the other three variants determined by side and tA. If $A$ is an m×n matrix, then, where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. See the documentation on factorize for more information. Anna Julia Cooper Intersectionality Since Crenshaw and Collins Concept taken feminist scholarship by storm Applied across a wide range of intersections Intersectionality applies to all of us We all experience a combination of privilege and oppression •gender •race •sexuality •class •age •ability •nation •religion See also normalize! Rank-2k update of the Hermitian matrix C as alpha*A*B' + alpha*B*A' + beta*C or alpha*A'*B + alpha*B'*A + beta*C according to trans. Generically sized uniform scaling operator defined as a scalar times the identity operator, λ*I. scale contains information about the scaling/permutations performed. Returns the vector or matrix X, overwriting B in-place. Multiplying a matrix by the identity matrix I (that's the capital letter "eye") doesn't change anything, just like multiplying a number by 1 doesn't change anything. If uplo = L, the lower triangle of A is used. Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor. If A has nonpositive eigenvalues, a nonprincipal matrix function is returned whenever possible. one(A*A') or one(A'*A) does the trick but is of course not what I want. Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently. Depending on side or trans the multiplication can be left-sided (side = L, Q*C) or right-sided (side = R, C*Q) and Q can be unmodified (trans = N), transposed (trans = T), or conjugate transposed (trans = C). Returns A. Rank-k update of the symmetric matrix C as alpha*A*transpose(A) + beta*C or alpha*transpose(A)*A + beta*C according to trans. The individual components of the decomposition F can be retrieved via property accessors: Iterating the decomposition produces the components Q, R, and if extant p. The following functions are available for the QR objects: inv, size, and \. Solves the equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization computed by gttrf!. Return the solution to A*x = b or one of the other two variants determined by tA and ul. This is equivalent to norm. If irange is not 1:n, where n is the dimension of A, then the returned factorization will be a truncated factorization. If compq = V the Schur vectors Q are updated. The identity operator I is defined as a constant and is an instance of UniformScaling. (The kth eigenvector can be obtained from the slice M[:, k].). Return the updated C. Return alpha*A*B or alpha*B*A according to side. Transforming rows of DataFrame ... transform the data from DataFrame to a value of a standard Matrix type available in Julia. tau contains scalars which parameterize the elementary reflectors of the factorization. Here is the solution I came up with. Test that a factorization of a matrix succeeded. Use norm to compute the Frobenius norm. If n and incx are not provided, they assume default values of n=length(dx) and incx=stride1(dx). = \prod_{j=1}^{b} (I - V_j T_j V_j^T)\], \[\|A\|_p = \left( \sum_{i=1}^n | a_i | ^p \right)^{1/p}\], \[\|A\|_1 = \max_{1 ≤ j ≤ n} \sum_{i=1}^m | a_{ij} |\], \[\|A\|_\infty = \max_{1 ≤ i ≤ m} \sum _{j=1}^n | a_{ij} |\], \[\kappa_S(M, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ See QRCompactWY. If range = A, all the eigenvalues are found. If jobvl = N, the left eigenvectors of A aren't computed. A is overwritten with its inverse. \[Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T).\], \[Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T) If uplo = U, the upper half of A is stored. Returns alpha*A*B or one of the other three variants determined by side and tA. requires at least Julia 1.3. If isgn = -1, the equation A * X - X * B = scale * C is solved. Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation of CC only uses O(n^2) operations. Overwrite b with the solution to A*x = b or one of the other two variants determined by tA and ul. Anna Julia Cooper Intersectionality Since Crenshaw and Collins Concept taken feminist scholarship by storm Applied across a wide range of intersections Intersectionality applies to all of us We all experience a combination of privilege and oppression •gender •race •sexuality •class •age •ability •nation •religion Notes Phys. Update vector y as alpha*A*x + beta*y where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. If jobu, jobv or jobq is N, that matrix is not computed. Solves A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) for (upper if uplo = U, lower if uplo = L) triangular matrix A. Solves the linear equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. Only the ul triangle of A is used. Comparing data frames for identity. Returns the eigenvalues of A. the 2nd to 8th eigenvalues. Finds the singular value decomposition of A, A = U * S * V'. Finds the LU factorization of a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal. No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. For real vectors v and w, the Kronecker product is related to the outer product by kron(v,w) == vec(w * transpose(v)) or w * transpose(v) == reshape(kron(v,w), (length(w), length(v))). A is overwritten by its Bunch-Kaufman factorization. Arrays can be used for storing vectors and matrices. (A), whereas norm(A, -Inf) returns the smallest. If A is complex symmetric then U' and L' denote the unconjugated transposes, i.e. If jobu = N, no columns of U are computed. T is a $n_b$-by-$\min(m,n)$ matrix as described above. Matrix factorizations (a.k.a. jobu and jobvt can't both be O. ipiv is the vector of pivots returned from gbtrf!. Examples. D is the diagonal of A and E is the off-diagonal. Julia automatically decides the data type of the matrix by analyzing the values assigned to it. The message that appears is: Warning: `eye(m::Integer)` has been deprecated in favor of `I` and `Matrix` constructors. tau must have length greater than or equal to the smallest dimension of A. Compute the LQ factorization of A, A = LQ. If transa = N, A is not modified. Return the generalized singular values from the generalized singular value decomposition of A and B. Computes the Bunch-Kaufman factorization of a symmetric matrix A. Returns the eigenvalues in W, the right eigenvectors in VR, and the left eigenvectors in VL. C is overwritten. If uplo = U, the upper half of A is stored. Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the computation of CC only uses O(n^2) operations. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. It is ignored when blocksize > minimum(size(A)). Matrix factorization type of the LDLt factorization of a real SymTridiagonal matrix S such that S = L*Diagonal(d)*L', where L is a UnitLowerTriangular matrix and d is a vector. A is assumed to be symmetric. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. Uses the output of gelqf!. A is overwritten with its LU factorization and B is overwritten with the solution X. ipiv contains the pivoting information for the LU factorization of A. Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B for square A. Modifies the matrix/vector B in place with the solution. F.Q, F.Z, and tau must have length greater than or equal to Slower unless A is itself (. Within the cycle error, each component-wise Algorithms are implemented for h \ B in-place [ AH16_6 ] )... A BLAS function has 4 methods defined, one each for Float64, Float32, and... Elements for each triangular matrix A singular vectors in iq X + X * B the! Not representable by the element type of eigen, the left eigenvectors of A matrix encounters! Weird numpy calls arising from the kth superdiagonal supported in Julia programs examples! May only implement norm ( A ) ), where op is determined by and. 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