gels! See also triu. If A is balanced with gebal! Similarly for transb and B. (The kth generalized eigenvector can be obtained from the slice F.vectors[:, k].) For matrices or vectors $A$ and $B$, calculates $Aᵀ / Bᵀ$. Modifies V in-place. Only the ul triangle of A is used. Just left values are numbered from 0. The input factorization C is updated in place such that on exit C == CC. Compute the cross product of two 3-vectors. which = 'LM': Eigenvalues with largest magnitude (eigs, eigsh), that is, largest eigenvalues in the euclidean norm of complex numbers.. which = 'SM': Eigenvalues with smallest … Construct an array by repeating the entries of A. The flop rate of the entire parallel computer is returned. * X = B (trans = T), or A' * X = B (trans = C) using the LU factorization computed by gttrf!. This returns a 5×5 Bidiagonal{Float64}, which can now be passed to other linear algebra functions (e.g. (The kth eigenvector can be obtained from the slice M[:, k].). Computes the generalized eigenvalue decomposition of A and B, returning a GeneralizedEigen factorization object F which contains the generalized eigenvalues in F.values and the generalized eigenvectors in the columns of the matrix F.vectors. Learn more. If job = O, A is overwritten with the columns of (thin) U and the rows of (thin) V'. Millions of developers and companies build, ship, and maintain their software on GitHub — the largest and most advanced development platform in the world. Most of the time it completes in 0.2-0.3 seconds, but occasionally it takes 7-8 seconds (without changing a and b). Returns X. By convention, a vector can be multiplied by a matrix on its left (A * v) whereas a row vector can be multiplied by a matrix on its right (such that v.' * A = (A.' I have been planning to do it for a while. If the perm argument is nonempty, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD's default AMD ordering). The Jacobi-Davidson implementation is ready for use and can be applied to solving the (generalized) eigenvalue problem for non-Hermitian matrices. tau contains scalars which parameterize the elementary reflectors of the factorization. Returns the vector or matrix X, overwriting B in-place. Returns the LU factorization in-place and ipiv, the vector of pivots used. The left Schur vectors are returned in vsl and the right Schur vectors are returned in vsr. x*y*z*... calls this function with all arguments, i.e. is the same as svdfact, but modifies the arguments A and B in-place, instead of making copies. T is a square matrix with min(m,n) columns, whose upper triangular part gives the matrix $T$ above (the subdiagonal elements are ignored). Compute A \ B in-place and store the result in Y, returning the result. Log of absolute value of matrix determinant. That said, I won't be able to get to it until next week - so anyone who wants to take a crack should probably do it. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). In particular, norm(A, Inf) returns the largest value in abs(A), whereas norm(A, -Inf) returns the smallest. Julia provides bindings to ARPACK, which can be used to perform iterative solutions for eigensystems (using eigs) or singular value decompositions (using svds).. eigs calculates the eigenvalues and, optionally, eigenvectors of its input(s) using implicitly restarted Lanczos or Arnoldi iterations for real symmetric or general nonsymmetric matrices respectively. svds(A) is formally equivalent to calling eigs to perform implicitly restarted Lanczos tridiagonalization on the Hermitian matrix $\begin{pmatrix} 0 & A^\prime \\ A & 0 \end{pmatrix}$, whose eigenvalues are plus and minus the singular values of $A$. P is a pivoting matrix, represented by jpvt. A collection of functions has been implemented as a MatrixEquation.jl Julia package to solve several classes of Lyapunov, Sylvester and Riccati matrix equations. nconv: Number of converged singular values. If side = B, both sets are computed. The residual associated with $θ$ and its corresponding Ritz vector $v$ is defined as the norm $||Av - vθ||$. A is overwritten by Q. Returns matrix C which is modified in-place with the result of the multiplication. The sigma and which keywords interact: the description of eigenvalues searched for by which do not necessarily refer to the eigenvalue problem $Av = Bv\lambda$, but rather the linear operator constructed by the specification of the iteration mode implied by sigma. Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA. (real symmetric. The (quasi) triangular Schur factor can be obtained from the Schur object F with either F[:Schur] or F[:T] and the orthogonal/unitary Schur vectors can be obtained with F[:vectors] or F[:Z] such that A = F[:vectors]*F[:Schur]*F[:vectors]'. Matrix inverse. maxiter: Maximum number of iterations (default = 300). * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrf!. Kronecker tensor product of two vectors or two matrices. Computes the polar decomposition of a vector. If job = E, only the condition number for this cluster of eigenvalues is found. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. Converts a symmetric matrix A (which has been factorized into a triangular matrix) into two matrices L and D. If uplo = U, A is upper triangular. * v2 returns a scalar, but will otherwise behave similarly. If sense = V, reciprocal condition numbers are computed for the right eigenvectors only. Hi, any news on this? Note that these restrictions limit the input matrix A to be of dimension at least 2. tol: relative tolerance used in the convergence criterion for eigenvalues, similar to tol in the eigs(A) method for the ordinary eigenvalue problem, but effectively for the eigenvalues of $B^{-1} A$ instead of $A$. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. See also normalize and vecnorm. The default is to compute a thin factorization. Computes the Bunch-Kaufman factorization of a Hermitian matrix A. \kappa_S(M, x, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p\], \[e^A = \sum_{n=0}^{\infty} \frac{A^n}{n! Instead of returning a new vector as qr(v::AbstractVector), this function mutates the input vector v in place. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. C is overwritten. Compute the QR factorization of A, A = QR. ilo, ihi, A, and tau must correspond to the input/output to gehrd!. For general matrices, the complex Schur form (schur) is computed and the triangular algorithm is used on the triangular factor. If range = A, all the eigenvalues are found. Same as ldltfact, but saves space by overwriting the input A, instead of creating a copy. The sigma and which keywords interact: the description of eigenvalues searched for by which do not necessarily refer to the eigenvalues of A, but rather the linear operator constructed by the specification of the iteration mode implied by sigma. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. select determines which eigenvalues are in the cluster. Constructs an upper (isupper=true) or lower (isupper=false) bidiagonal matrix using the given diagonal (dv) and off-diagonal (ev) vectors. If transa = N, A is not modified. The eigenvalues are returned in w and the eigenvectors in Z. Computes the eigenvectors for a symmetric tridiagonal matrix with dv as diagonal and ev_in as off-diagonal. nb sets the block size and it must be between 1 and n, the second dimension of A. A may be represented as a subtype of AbstractArray, e.g., a sparse matrix, or any other type supporting the four methods size(A), eltype(A), A * vector, and A' * vector. Update a Cholesky factorization C with the vector v. If A = C[:U]'C[:U] then CC = cholfact(C[:U]'C[:U] + v*v') but the computation of CC only uses O(n^2) operations. C is overwritten. hessfact! When running in parallel, only 1 BLAS thread is used. If uplo = L, the lower triangle of A is used. U, S, V and Vt can be obtained from the factorization F with F[:U], F[:S], F[:V] and F[:Vt], such that A = U*diagm(S)*Vt. Returns A, containing the bidiagonal matrix B; d, containing the diagonal elements of B; e, containing the off-diagonal elements of B; tauq, containing the elementary reflectors representing Q; and taup, containing the elementary reflectors representing P. Compute the LQ factorization of A, A = LQ. JacobiDavidson.jl. If range = I, the eigenvalues with indices between il and iu are found. If diag = U, all diagonal elements of A are one. Note that these restrictions limit the input matrix A to be of dimension at least 2. which: type of eigenvalues to compute. If uplo = U the upper Cholesky decomposition of A was computed. For matrices, the matrix norm induced by the vector p-norm is used, where valid values of p are 1, 2, or Inf. If symmetric is true, A is assumed to be symmetric. If you want to overload these operations for your own types, then it is useful to know the names of these functions. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. The solution is returned in B. Solves the linear equation A * X = B where A is a square matrix using the LU factorization of A. F[:D2] is a P-by-(K+L) matrix whose top right L-by-L block is diagonal. This handout will give you an Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If range = A, all the eigenvalues are found. Computes eigenvalues d of A using implicitly restarted Lanczos or Arnoldi iterations for real symmetric or general nonsymmetric matrices respectively. Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (schur) and then the complex square root of the triangular factor. Returns alpha*A*x or alpha*A'*x according to trans. Nonlinear eigenvalue problems (NEPs) arise in many fields of science and engineering. A is overwritten with its inverse. A (non-zero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies the linear equation = where λ is a scalar, termed the eigenvalue corresponding to v.That is, the eigenvectors are the vectors that the linear transformation A merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. Those BLAS functions that overwrite one of the input arrays have names ending in '!'. In julia they are numbered from 1. I have two huge sparse matrices A and B generated by FreeFem++. This format should not to be confused with the older WY representation [Bischof1987]. Overwrites B with the solution X and returns it. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A. Compute the blocked QR factorization of A, A = QR. the unique matrix $X$ such that $e^X = A$ and $-\pi < Im(\lambda) < \pi$ for all the eigenvalues $\lambda$ of $X$. Finds the singular value decomposition of A, A = U * S * V'. Compute the Bunch-Kaufman [Bunch1977] factorization of a symmetric or Hermitian matrix A and return a BunchKaufman object. If itype = 2, the problem to solve is A * B * x = lambda * x. 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 diagm(R)*B (if trans = N and equed = R,B) or diagm(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. Computes the Generalized Schur (or QZ) factorization of the matrices A and B. Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. D is the diagonal of A and E is the off-diagonal. By default, if no arguments are specified, it multiplies a matrix of size n x n, where n = 2000. Computes Q * C (trans = N), Q.' Compute a convenient factorization of A, based upon the type of the input matrix. For more information, see [issue8859], [B96], [S84], [KY88]. ritzvec: Returns the Ritz vectors v (eigenvectors) if true, v0: starting vector from which to start the iterations. If jobvt = O, A is overwritten with the rows of (thin) V'. For matrices or vectors $A$ and $B$, calculates $Aᴴ / Bᴴ$. It may have length n (the second dimension of A), or 0. svd: An SVD object containing the left singular vectors, the requested values, and the right singular vectors. Sign in For details of how the errors in the computed eigenvalues are estimated, see: B. N. Parlett, "The Symmetric Eigenvalue Problem", SIAM: Philadelphia, 2/e (1998), Ch. Wrapper around svdfact extracting all parts of the factorization to a tuple. Compute the generalized SVD of A and B, returning a GeneralizedSVD factorization object F, such that A = F[:U]*F[:D1]*F[:R0]*F[:Q]' and B = F[:V]*F[:D2]*F[:R0]*F[:Q]'. Most of the below functionality described in the core MATLAB Mathematics documentation has equivalent, often identical, functionality (more often that not with the same syntax) described in the Base.Mathematics section of the Julia manual. If compq = N, only the singular values are found. Computes the inverse of a Hermitian matrix A using the results of sytrf!. Confirm if a specific wavefunction is an eigenfunction of a specific operation and extract the corresponding obserable (the eigenvalue) To recognize that the Schrödinger equation, just like all measurable, is also an eigenvalue problem with the eigenvalue ascribed to total energy; Identity and manipulate several common quantum mechanical operators A lazy-view wrapper of an AbstractArray, taking the elementwise complex conjugate. Mathematics []. An object of type UniformScaling, representing an identity matrix of any size. If isgn = 1, the equation A * X + X * B = scale * C is solved. Returns U, S, and Vt, where S are the singular values of A. If jobvr = N, the right eigenvectors aren't computed. dA determines if the diagonal values are read or are assumed to be all ones. * A)[F[:p], F[:q]]. job can be one of N (A will not be permuted or scaled), P (A will only be permuted), S (A will only be scaled), or B (A will be both permuted and scaled). * X = B (trans = T), A' * X = B (trans = C) for side = L, or the equivalent equations a right-handed side = R X * A after computing X using trtrs!. Returns A*x. Solves the linear equation A * X = B (trans = N), A.' Finds the eigensystem of A with matrix balancing. If norm = O or 1, the condition number is found in the one norm. Equivalent to log(det(M)), but may provide increased accuracy and/or speed. factors, as in the QR type, is an m×n matrix. on A. Only the ul triangle of A is used. Returns the upper triangle of M starting from the kth superdiagonal, overwriting M in the process. Set the number of threads the BLAS library should use. The entries of F[:D1] and F[:D2] are related, as explained in the LAPACK documentation for the generalized SVD and the xGGSVD3 routine which is called underneath (in LAPACK 3.6.0 and newer). A Ritz value $θ$ is considered converged when its associated residual is less than or equal to the product of tol and $max(ɛ^{2/3}, |θ|)$, where ɛ = eps(real(eltype(A)))/2 is LAPACK's machine epsilon. If jobu = A, all the columns of U are computed. * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a LQ factorization of A computed using gelqf!. τ is a vector of length min(m,n) containing the coefficients $au_i$. Generalized Eigenvalues¶ It is sometimes useful to consider the generalized eigenvalue problem, which, for given matrices $ A $ and $ B $, seeks generalized eigenvalues $ \lambda $ and eigenvectors $ v $ such that $$ A v = \lambda B v $$ This can be solved in Julia via eigen(A, B). Computes Q * C (trans = N), Q.' (A,b) scales each column i of A by b[i] (similar to A*Diagonal(b)), while scale! If uplo = U, the upper half of A is stored. doi:10.1137/120885991. I encountered a strange issue when solving the generalized eigenvalue problem in Julia 0.6 (version on mac os, but also on 0.5 and 0.6 in Juliabox). Explicitly finds the matrix Q of a QL factorization after calling geqlf! (The kth eigenvector can be obtained from the slice M[:, k]. The following functions are available for the QR objects: inv, size, and \. Matrix trace. Construct a Symmetric view of the upper (if uplo = :U) or lower (if uplo = :L) triangle of the matrix A. Computes the generalized eigenvalues of A and B. Equivalent to (log(abs(det(M))), sign(det(M))), but may provide increased accuracy and/or speed. Solves the Sylvester matrix equation A * X +/- X * B = scale*C where A and B are both quasi-upper triangular. Only the uplo triangle of A is used. The result is of type Bidiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). If A is a matrix and p=2, then this is equivalent to the Frobenius norm. Same as ordschur but overwrites the factorization F. Same as ordschur but overwrites the input arguments. Copy n elements of array X with stride incx to array Y with stride incy. Computes the least norm solution of A * X = B by finding the SVD factorization of A, then dividing-and-conquering the problem. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. Finds the reciprocal condition number of matrix A. Note that Y must not be aliased with either A or B. Finds the generalized eigendecomposition of A and B. Awad H. Al-Mohy and Nicholas J. Higham, "Improved inverse scaling and squaring algorithms for the matrix logarithm", SIAM Journal on Scientific Computing, 34(4), 2012, C153-C169. If A has no negative real eigenvalues, compute the principal matrix square root of A, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. * X =B, or A' * X = B using a QR or LQ factorization. No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. The eigenvalues of A can be obtained with F[:values]. I wanted to solve a generalized eigenvalue problem but none of the crates I could find at the time supported anything but calling BLAS directly, while it was trivial to do it with Julia. No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions. Solve the equation AB * X = B. trans determines the orientation of AB. Can optionally also compute the product Q' * C. Returns the singular values in d, and the matrix C overwritten with Q' * C. Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal using a divide and conqueq method. The singular values in S are sorted in descending order. Rank-k update of the Hermitian matrix C as alpha*A*A' + beta*C or alpha*A'*A + beta*C according to trans. Explicitly finds the matrix Q of a QR factorization after calling geqrf! Otherwise they should be ilo = 1 and ihi = size(A,2). If jobu = S, the columns of (thin) U are computed and returned separately. A is overwritten by its Bunch-Kaufman factorization. If ritzvec = false, the left and right singular vectors will be empty. If uplo = U, the upper half of A is stored. usually also require fine-grained control over the factorization of A. A is overwritten by its inverse and returned. If isgn = -1, the equation A * X - X * B = scale * C is solved. w_in specifies the input eigenvalues for which to find corresponding eigenvectors. Uses the output of geqrf!. A is the LU factorization from getrf!, with ipiv the pivoting information. If [vl, vu] does not contain all eigenvalues of A, then the returned factorization will be a truncated factorization. if A == A.'). irange is a range of eigenvalue indices to search for - for instance, the 2nd to 8th eigenvalues. If A is symmetric or Hermitian, its eigendecomposition (eigfact) is used to compute the square root. Left division operator: multiplication of y by the inverse of x on the left. Only the ul triangle of A is used. If B is nearly symmetric positive definite, then consider using B = (B+B')/2 to make B symmetric before calling eigs. Computes the polar decomposition of a vector. Solves the equation A * X = B where A is a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal. Same as schurfact but uses the input matrices A and B as workspace. ), Computes the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F[:values] and the eigenvectors in the columns of the matrix F[:vectors]. Finds the singular value decomposition of A, A = U * S * V', using a divide and conquer approach. Construct a symmetric tridiagonal matrix from the diagonal and first sub/super-diagonal, respectively. Returns op(A)*b, where op is determined by tA. In the latter case, the optional argument uplo may be :L for using the lower part or :U for the upper part of A. Multiplies the matrix C by Q from the transformation supplied by tzrzf!. Computes the least norm solution of A * X = B by finding the full QR factorization of A, then dividing-and-conquering the problem. Only the uplo triangle of A is used. See also lq. B is overwritten by the solution X. If compq = N they are not modified. B is overwritten with the solution X. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. scale contains information about the scaling/permutations performed. Returns A*B or B*A according to side. If howmny = S, only the eigenvectors corresponding to the values in select are computed. If uplo = L, the lower half is stored. Explicitly finds the matrix Q of a RQ factorization after calling gerqf! Returns the updated B. '*A + beta*C according to trans. Only works for real types. This is accomplished through the keyword which.The following values of which are available:. Computes the Bunch-Kaufman factorization of a symmetric matrix A. For matrices M with floating point elements, it is convenient to compute the pseudoinverse by inverting only singular values above a given threshold, tol. For numbers, return $\left( |x|^p \right) ^{1/p}$. A Q matrix can be converted into a regular matrix with full which has a named argument thin. Compute the LU factorization of a sparse matrix A. A is overwritten by its Bunch-Kaufman factorization. If thin=true (default), a thin SVD is returned. For matrices or vectors $A$ and $B$, calculates $A$ \ $Bᴴ$. Solves the equation A * X = B for a symmetric matrix A using the results of sytrf!. This function requires LAPACK 3.6.0. Transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). The vector v is destroyed during the computation. If uplo = U, A is upper triangular. iblock_in specifies the submatrices corresponding to the eigenvalues in w_in. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. A is overwritten by its inverse. Nicholas J. Higham, "The squaring and scaling method for the matrix exponential revisited", SIAM Journal on Matrix Analysis and Applications, 26(4), 2005, 1179-1193. doi:10.1137/090768539. If jobvr = N, the right eigenvectors of A aren't computed. If diag = N, A has non-unit diagonal elements. If order = E, they are ordered across all the blocks. dA determines if the diagonal values are read or are assumed to be all ones. If diag = U, all diagonal elements of A are one. Normalize the vector v so that its p-norm equals unity, i.e. Currently only implemented for sparse matrices. If uplo = L, the lower half is stored. for integer types. isplit_in specifies the splitting points between the submatrix blocks. If sense = E, reciprocal condition numbers are computed for the eigenvalues only. A is overwritten by Q. Computes Q * C (trans = N), Q.' Returns alpha*A*x or alpha*A'x according to tA. Overwrite Y with a*X + Y, where a is a scalar. Computes the inverse of a symmetric matrix A using the results of sytrf!. alpha is a scalar. A is overwritten by Q. The function calls the C library SPQR and a few additional functions from the library are wrapped but not exported. Computes the LDLt factorization of a positive-definite tridiagonal matrix with D as diagonal and E as off-diagonal. The eigenvalues of A are returned in the vector λ. Reorders the Schur factorization F of a matrix A = Z*T*Z' according to the logical array select returning the reordered factorization F object. If job = V, only the condition number for the invariant subspace is found. It differs from a 1×n-sized matrix by the facts that its transpose returns a vector and the inner product v1.' Successfully merging a pull request may close this issue. * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QL factorization of A computed using geqlf!. alpha and beta are scalars. 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, A.' If range = I, the eigenvalues with indices between il and iu are found. Compute the Cholesky factorization of a dense symmetric positive definite matrix A and return a Cholesky factorization. A lazy-view wrapper of an AbstractVector, which turns a length-n vector into a 1×n shaped row vector and represents the transpose of a vector (the elements are also transposed recursively). For matrices or vectors $A$ and $B$, calculates $Aᴴ$ \ $Bᴴ$. ... Compute the eigenvalue decomposition of A and return an Eigen object. If info = 0, the factorization succeeded. I have a Hermitian matrix H and a Hermitian and positive definite matrix S. I want the solutions to the following generalized eigenvalue problem Hx=\lambda Sx and want x to be properly normalized. peakflops computes the peak flop rate of the computer by using double precision gemm!. Returns the smallest eigenvalue of A. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals, and alpha is a scalar. eigs returns the nev requested eigenvalues in d, the corresponding Ritz vectors v (only if ritzvec=true), the number of converged eigenvalues nconv, the number of iterations niter and the number of matrix vector multiplications nmult, as well as the final residual vector resid. Multiplication with respect to either thin or full Q is allowed, i.e. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If B is provided, the generalized eigen-problem is solved. privacy statement. C is overwritten. Scale an array A by a scalar b overwriting A in-place. tau stores the elementary reflectors. Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. Same as eigvals, but saves space by overwriting the input A (and B), instead of creating copies. svd is a wrapper around svdfact, extracting all parts of the SVD factorization to a tuple. jpvt must have length greater than or equal to n if A is an (m x n) matrix. Many BLAS functions accept arguments that determine whether to transpose an argument (trans), which triangle of a matrix to reference (uplo or ul), whether the diagonal of a triangular matrix can be assumed to be all ones (dA) or which side of a matrix multiplication the input argument belongs on (side). to your account. Construct a Bidiagonal matrix from the main diagonal of A and its first super- (if isupper=true) or sub-diagonal (if isupper=false). The following keyword arguments are supported: nev: Number of eigenvalues. If the keyword argument parallel is set to true, peakflops is run in parallel on all the worker processors. An implementation of Jacobi-Davidson in Julia. For matrices or vectors $A$ and $B$, calculates $A$ \ $Bᵀ$. If fact = F and equed = C or B the elements of C must all be positive. Returns the lower triangle of M starting from the kth superdiagonal. Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal, respectively. The Schrödinger Equation gives the solutions to the problem and is an eigenvalue problem. Only the ul triangle of A is used. Returns a matrix M whose columns are the generalized eigenvectors of A and B. If uplo = L, the lower half is stored. Test whether a matrix is lower triangular. Compute the (pivoted) QR factorization of A such that either A = Q*R or A[:,p] = Q*R. Also see qrfact. Only the ul triangle of A is used. If false, omit the singular vectors. If uplo = L, the lower half is stored. For any iterable containers x and y (including arrays of any dimension) of numbers (or any element type for which dot is defined), compute the Euclidean dot product (the sum of dot(x[i],y[i])) as if they were vectors.

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