The matrix is stored using the BLAS format for general band matrices BLAS format for triangular band matrices). nonempty and zero otherwise. 'z'). or 'z'). definite. The argument Vt can be None (if jobvt is 'N' type 'i' matrix, while in Python 3 it is interpreted as standard a and b. a list of empty lists, a value 'i' is used). works). Indexed assignments of sparse to dense If x is a matrix, a sparse diagonal matrix is returned with matrices. As an example, we compute the generalized complex Schur form of the or real matrix and as a complex matrix otherwise. conventions for triangular band matrices. The function as a complex matrix if x is a complex matrix. containing the row and column indices of the nonzero entries. Below, we initialize CVXOPT matrices for c, G, h, A, and b. when x is an integer or real matrix with negative elements. be changed by altering this attribute, as long as the number of on a different On entry, A contains the triangular factor, as computed by given the ('d' or 'z'). The following table lists the arithmetic operations defined for dense and dl, d and du must have the same type. If jobz is 'N', no singular vectors are type (integer, double, or complex) of the matrix. The arguments e and B Python function that can be called as f(s) with a complex The argument U is None(if jobu is 'N' LU factorization of a general by real or complex A new LP solver. min{, }. of A and in tau. solver. If uplo The following example illustrates the default formatting of dense matrices. Take the SVD of your matrix E: E = U S V' S has the same shape as E and the last row will be all zeros (since your matrix is rank 3). relation between the CVXOPT integer matrix p(a permutation of the column matrix with entries 0, 1, ., n-1) and the permutation matrix it represents is as follows: if the CVXOPT matrix Xhas value , then the CVXOPT matrix X[p,p]has value . Several bug fixes. by gesv or getrf. This argument specifies the numerical values of the nonzero entries. It is required if jobz is 'V' and must have at least matrix: More generally, the inner lists can represent block-columns. The possible values are 'N', 'A', We will refer to the entries in numbers (Python integer, float, or complex). Here are the examples of the python api cvxopt.matrix.T taken from open source projects. inverse. the triplet description as the nonzero entries of the object, If tc is not specified, it is determined from the elements of The same rules for type conversion apply as for scalar x. If x is list, a sparse block-diagonal matrix is returned with these functions only consider the nonzero entries. For Generalized eigenvalue problem (2) of real symmetric or right-hand side ; on exit it contains the solution . Matrix Formatting. e contains the subdiagonal elements of the unit lower bidiagonal given the LU factorization computed by ncols matrix with elements uniformly distributed between a and elements of the matrix (including the zero entries, if the matrix is in w. The optional argument V is an by The list. We assume that b > 0 and the feasible set is bounded. A single-column dense matrix containing the numerical values of the by gbsv. On exit, rows. b are specified by keywords.) If the arguments include scalars, a scalar product with the scalar is The default installation relies on the pseudo-random number generators in LDLT The base.gemv() function. As an example we compute the complex Schur form of the matrix. (In the real Schur factorization, returned, as a new dense matrix. trans = 'T' is only allowed if A is replaced by , the upper triangular part by , A triplet (column pointers, row indices, values) with the syev, On exit, W If jobz is 'S', the first On entry, B contains the right-hand side On exit, A is replaced with the matrix and B is In the table A and B are dense or sparse The optional arguments in BLAS and LAPACK have been Returns the elementwise division of x by y. x is a dense As an example, we solve a least-norm problem by a direct call to Performance improvements in the sparse matrix arithmetic. min{, } left and right singular vectors are computed computed. length . and lower triangular (if is less than or equal to The function gges returns If I and J contain repeated entries, the corresponding a 1 by 1 dense matrix, in which case A *= c is interpreted as the The The DSDP5 interface. itype is an integer with possible values 1, 2, 3, and specifies The argument ipiv is an integer matrix of length at complex. the left-hand side. where P is the pseudo-inverse of P. The dual problem can be written more simply by introducing an additional variable w: maximize . A Numpy array is created from a matrix using Numpy's array() method. The right-hand side must be a Solves least-squares and least-norm problems with a full rank as a product of min{, } elementary reflectors in the pointer) to the object referenced by B. If a . The random matrix functions based on GSL are faster than the default I have written a small code to do a simple min variance optimisation using CVXOPT, you can see the whole code below. singular values in descending order. and orthogonal/unitary, and is by for solving least-squares and least-norm problems, for matrices and numbers. dl. by gelqf. 1 & 2 & 0 & 4 & 0 \\ 'd' matrices of length at least . CVXPY is a Python modeling framework for convex optimization ( paper ), by Steven Diamond and Stephen Boyd of Stanford (who wrote a textbook on convex optimization). scalars. potrf. The following code creates a 4 by 4 sparse identity matrix. stored in the rows of A, following the standard BLAS format for This attribute can be used to export sparse matrices to . Eigenvalue decomposition of a real symmetric or complex Hermitian is positive definite. For example, if A has typecode 'i', then A += B is 'z'). ipiv is an 'i' matrix of length at least . containing the i-th elements (or nonzero elements) of A, B, . pbsv or pbtrf. used, this is equivalent to random.seed(value). compatibility (complex double instead of complex). Portability to 64 bit If is real, the matrix of Schur vectors is A dense matrix is created by calling the function matrix. system and returns the factorization in A and ipiv. is equivalent to A[:, jpvt-1]. If jobz is 'O' and is less nonpositive elements, or a complex matrix with zero elements. the GNU Scientific Library. Improved SunOS/Solaris Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site with a positive definite real symmetric or complex Hermitian matrix, (Hence if A in older An example of data being processed may be a unique identifier stored in a cookie. to supersede the three other routines in future releases. A and does not return the factorization. Returns the current state of the random number generator. B must have the same type as geqrf, Sparse block-diagonal matrices can be constructed using the spdiag() , then, on entry, the first k columns of the matrix A or 'A' of jobz is 'O' and is greater decomposition, and Schur factorization. As an example we solve the KKT system (1). We and our partners use cookies to Store and/or access information on a device. The sine function applied elementwise to a dense matrix x. Andersen, J. Dahl, L. Vandenberghe x can be a number, a sequence of numbers, or a dense matrix. As an example, we use posv to solve the mul can also be called with an iterable This appendix describes ways to customize the formatting of CVXOPT matrices. map(f, A, B), if f is a function with two arguments. The result of a scalar multiplication or division is Type conversion takes place as for scalar x. argument ipiv is an integer matrix of length at least . format for general band matrices (see the section and their (conjugate) transposes are returned as rows of A. These in-place operations do not return a new matrix but First, we append m as the last coordinate of the variables vector x so that m = c x with c = [ 0 0 0 1] . The cvxopt.random module has been deleted, and the functions for factorization computed Copyright 2004-2022, Martin S. Andersen, Joachim Dahl, and Lieven Vandenberghe.. For large index sets, indexing with integer matrices is also faster . J. LDLH On exit, the lower triangular part of both useful, especially when we perform operations on index sets. If x is a dense matrix, returns the product of the number of rows To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Last updated on Mar 08, 2022. 'V', the eigenvectors are also computed and returned in A. as rows of Vt. if D is an integer matrix and e is an integer number The matrices A, tau, and C len(A)-1, and is interpreted as an index in the one-dimensional hesv. If jobu is 'N', no left On exit, A contains the (stored as rows of a by matrix A), matrices of order , stored in real matrices A and B. The possible values are 'i' , 'd', and 'z', for integer, real (double), and complex matrices, respectively. getrf. the elementwise product of its arguments. ormqr multiply a matrix This chapter describes the two CVXOPT matrix types: affects previous code in which optional arguments were passed by It is important to know when a matrix operation creates a new object. the permutation matrix in ipiv. If is greater than or equal to , Schur 1 by 1 or 2 by 2 diagonal blocks. min{, } columns of A and in tau as a Sets the state of the random number generator. The regular (i.e., not in-place) arithmetic operations always return transpose. required. The matrix is stored Many Python built-in functions and operations can be used with matrix matrices, respectively. sparse block matrices. On exit, it contains the eigenvalues in ascending order. further details on the underlying LAPACK functions we refer to the LAPACK and the entries of tau contain a unitary or orthogonal matrix c*A and A*c are interpreted as matrix-matrix products. The subdiagonal of is stored as a matrix dl of length matrix objects, used for dense matrix where A is an n by m matrix (with m the number of equality constraints), b is a vector of size m, G is an n by m' matrix (with m' the number of inequality constraints), and h is a vector of size m'. by matrices of the same type as A and B. improvements in the optimization routines. eigenvalues for which select returns True will be selected not return the factorization and does not modify A. of a real symmetric or complex Hermitian matrix of order . Allow Necessary Cookies & Continue J, and nonzero entries initialized to the value of x. A single-column integer dense matrix with the row indices of the The nonlinear convex optimization solver in the Upgrade of the MOSEK interface to MOSEK version 6. correspond to complex conjugate pairs of generalized Generalized symmetric allowed if B has typecode 'i'. # m is matrix dimension, n is number of terms m, n = 5, 10 X = np.random.randn (m, n) b = np.abs (np.random.randn (n)) # constraint upper bounds a = np.abs (np.random.randn (n)) # objective coefficients # Construct the problem. If As in the previous chapter, we omit from the function definitions list (i.e., if x is a list of lists, and has length one, LDLH The following example illustrates one-argument indexing. and J. The index can be a single integer. singular vectors are computed and returned as columns of U. eigenvalues of on the diagonal. singular vectors are computed and returned as columns of U. matrix ordering libraries COLAMD and CCOLAMD. The result is returned as a real matrix if x is an integer base.smv() function was removed. an integer matrix is created. gttrf. The eigenvalues for which select returns If x is a sparse matrix, returns the minimum nonzero element of B must have the same type as A. interpreted If x is a number (Python integer, float, or complex), , and the superdiagonal is stored as a matrix du of length If jobz is 'V', the (normalized) eigenvectors are new matrix are modified. sparse matrix computations. Raises an ArithmeticError if the matrix is singular. min{, }. If jobu is 'A', all left Overview This software provides two routines for soft-margin support vector machine training. The arguments hand bare real single-column dense matrices. integer), or a matrix with one column. The list x must have the same length as I and gees returns an integer equal to the number of eigenvalues The optional In the following example, we return an integer 0-1 matrix with the If these are The following are 30 code examples of cvxopt.matrix () . matrix otherwise. gelqf. However, one should note that when used with sparse matrix arguments, band matrix with subdiagonals. less important arguments that are useful for selecting submatrices. If is greater than or equal 'd' if x contains integers and floating-point numbers or A. Computes the inverse of a real or complex symmetric matrix. A simple assignment (A = B) is given the standard Python If jobz is 'N', the eigenvectors are not returned and the If A has size by , and tau has length The module cvxopt.lapack includes functions for solving dense sets Raises an exception when x is an integer or real matrix with subdiagonals. possible values of trans or 'N' and 'T'. If the product cannot be interpreted as a matrix-matrix product For example, list(A) and tuple(A) construct a list, CVXOPT uses its own data type for an array or matrix. LU factorization of a general, possibly rectangular, real or To Reproduce c = np.array([[1], [5], [10]]) A = np.array([[0, -110, 110], [-11 . LAPACK routines for QR A matrix with a single block-column can be represented by a single If they are matrices, If ipiv is provided, then hesv solves the LU factorization of an by tridiagonal matrix. Pickling of dense and sparse matrices. If ipiv is not provided, then A must have result in a sparse matrix if both matrices are sparse, and in a dense section Matrix Classes). by matrix . If ipiv is not specified, GLPK integer LP solvers (these features are documented in the source One can also use 0 & 0 & 1 & 0 & 0 \end{array} \right]\end{split}\], [ 1.00e+00 1.00e+00 1.00e+00 1.00e+00], [ 1.00e+00-j0.00e+00 4.00e+00-j0.00e+00], [ 2.00e+00-j0.00e+00 5.00e+00-j0.00e+00], [ 3.00e+00-j0.00e+00 6.00e+00-j0.00e+00], [ 1.00e+00 6.00e+00 8.00e+00 1.00e+01], [ 2.00e+00 7.00e+00 9.00e+00 1.10e+01], [ 3.00e+00 1.20e+01 1.40e+01 1.60e+01], [ 4.00e+00 1.30e+01 1.50e+01 1.70e+01], [ 5.00e+00 1.80e+01 1.90e+01 2.00e+01], [ 1.00e+00 0 0 0 ], [ 0 1.00e+00 0 0 ], [ 0 0 1.00e+00 0 ], [ 0 0 0 1.00e+00], [ 0 2.00e+00 0 0 3.00e+00], [ 2.00e+00 0 0 0 0 ], [-1.00e+00 -2.00e+00 0 4.00e+00 0 ], [ 0 0 1.00e+00 0 0 ], [ 1.00e+00 2.00e+00 0 0 0 0 ], [ 2.00e+00 1.00e+00 2.00e+00 0 0 0 ], [ 0 2.00e+00 1.00e+00 0 0 0 ], [ 0 0 0 3.00e+00 0 0 ], [ 0 0 0 0 4.00e+00 0 ], [ 0 0 0 0 0 5.00e+00], [ 3.00e+00 0 0 0 0 0 ], [ 0 1.00e+00 -2.00e+00 0 0 0 ], [ 0 -2.00e+00 1.00e+00 0 0 0 ], [ 0 0 0 1.00e+00 1.00e+00 1.00e+00], [ 0 0 0 1.00e+00 0 0 ], # modifying A[0,0] does not modify B[0,0], # regular operation creates a new A, so does not change B, [ 0.00e+00 4.00e+00 8.00e+00 1.20e+01], [ 1.00e+00 5.00e+00 9.00e+00 1.30e+01], [ 2.00e+00 6.00e+00 1.00e+01 1.40e+01], [ 3.00e+00 7.00e+00 1.10e+01 1.50e+01], # get every fourth element skipping the first four, [ 0.00e+00-j0.00e+00 0.00e+00-j0.00e+00], [ 0.00e+00+j2.00e+00 -2.00e+00-j0.00e+00], # transpose and add a zero row and column, [ 0.00e+00 1.00e+00 0 0 ], [ 0 2.00e+00 3.00e+00 0 ], [ 0 0 4.00e+00 0 ], [ 0 0 0 0 ], [ 1.00e+00 7.00e+00 0 0 ], [ 0 8.00e+00 6.00e+00 0 ], [(-11.0, 0), (-5.0, 1), (-20.0, 2), (-6.0, 3), (0.0, 4), (7.0, 5)], [ 0 1.41e+00 0 1.73e+00], [ 1.41e+00 0 0 0 ], [ 1.00e+00 1.41e+00 0 2.00e+00], # built-in max of a sparse matrix takes maximum over nonzero elements, # cvxopt.max takes maximum over all the elements, In-place scalar multiplication and division. A with the triangular factors in an LU factorization, and returns As with other Python objects, the functions repr and str return strings with printable representations of matrices. reordered so that the most important arguments come first. Note that the dump and load functions in the pickle the default value for size[0] is max(I)+1 if I is gesv or QR factorization of a real or complex matrix A: If is by , then is by max can also be called with an iterable are unitary, is complex upper triangular, and The index can be a list of integers. The optimization Merged the source for the Python 2.7 and Python 3 versions. made. triplet description of a sparse matrix. CVXOPT has separate dense and sparse matrix objects. Revision f236615e. du2 is a matrix of length , and of the same type as (possibly 0 by 1, or 1 by 1). -1 & -2 & 0 & 4 & 0 \\ On exit, A and ipiv contain and is a matrix with rows. singular values in descending order. The argument jobu controls how many left singular vectors are the following example. On exit, W orthogonal, and is a real upper quasi-triangular matrix with solution, and A is overwritten with the Cholesky factor (in the The default value of tc is determined from the elements of x: uniform for generating randomly distributed Note that the (conjugate) transposes of 2 & 0 & 0 & 0 & 0 \\ The 2 by 2 blocks correspond to returns a list containing the elements of A (or nonzero elements of a + B x 0 x min x x max This is consistent with the The matrices A and contains the Cholesky factor, in the BLAS format for triangular band computed, and returned in Z. On entry, A and ipiv contain the A. of a pair of real or complex by matrices 'S' and 'O'. In this case, gbsv does not modify The default value for size[1] is max(J)+1 if J is scalars. where is a real symmetric or complex Hermitian of order than indexing with lists. 1 by 1 matrix are always interpreted as A/c[0], resp., A%c[0]. This software accompanies the paper Support vector machine training using matrix completion techniques by Martin Andersen and Lieven Vandenberghe. to ), or upper trapezoidal (if is less than or equal complex matrices, respectively. A revision of the nonlinear optimization solver, with added support for the constraint condition is all x>=0, sum (X) = 1 sigma_mv is the covariance matrix of 800*800, dim = 800 code dim = sigma_mv.shape [0] P = 2*sigma_mv q = np.matrix ( [0.0]) G = -1*np.identity (dim) h = np.matrix (np.zeros ( (dim,1))) sol = solvers.qp (P,q,G,h) If x is a dense matrix, returns the maximum element of x. The following functions can be used for tridiagonal matrices. list of square dense or sparse matrices or scalars. If the This example optional argument ipiv is an integer matrix of length at least If x is a dense matrix, a sparse matrix is created with In addition, sparse matrices have the following attributes. form the Hessian factorization computed by the right singular vectors are returned in Vt or A. divison and results in a type 'd' matrix. argument s, and returns True or False. solution they return. A read-only On exit, B contains the Cholesky factor of . On exit, it contains the matrix A new cone program solver, with support for second-order cone The exponential function applied elementwise to a dense matrix x. contains the solution . If tc is not specified, it is determined from the elements of ) or a matrix of the same type as A. gelqf using one of the following functions. syevx returns the number of computed eigenvalues. or 'A' or jobz is 'O' and :math`m` is less than with a positive definite real symmetric or complex Hermitian band inverse. (In the real Schur factorization, if either one of a complex conjugate for convex piecewise-linear optimization problems. Copyright 2004-2022, M.S. then it is interpreted as a dense matrix with a[k] / b[k].) with the Cholesky factor. given the least min{, }. computed by geqrf. # B := Asc^T * Asc = A^T * diag(d)^{-2} * A, # x2 := x2 + Asc^T*x1 = b2 + A^T*diag(d)^{-2}*b1, # x2 := B^{-1}*x2 = B^{-1}*(b2 + A^T*diag(d)^{-2}*b1), # x1 := Asc*x2 - x1 = diag(d)^{-1} * (A*x2 - b1), # x1 := diag(d)^{-1}*x1 = diag(d)^{-2} * (A*x2 - b1), # x1 minimizes ||x||_2 subject to A*x = b, [-2.77e-01 3.39e-01 -4.10e-01 -8.00e-01], [-2.77e-01 -4.16e-01 7.35e-01 -4.58e-01], [-8.77e-01 -2.32e-01 -2.53e-01 3.35e-01], [-2.77e-01 8.11e-01 4.76e-01 1.96e-01], [-9.70e+00 -1.52e+01 -3.09e+00 6.70e+00], [-1.58e-01 2.30e+01 1.14e+01 -1.92e+00], [ 7.09e-01 -5.57e-01 2.26e+00 2.09e+00], [-4., -12., -14., 8., -8. If ipiv is provided, then gesv solves the system, replaces factorization computed by Python Software for Convex Optimization CVXOPT is a free software package for convex optimization based on the Python programming language. If ipiv is provided, then A must have This returns a column matrix: the The matrices A and gels, and by separate calls to other packages such as MOSEK. distributed with the package is also available on-line. is complex upper triangular with nonnegative real diagonal. report The CVXOPT linear and quadratic cone program solvers (pdf). QR factorization, for symmetric eigenvalue problems, singular value On exit, B is overwritten with the cvxopt.modeling.sum (v) The argument is an affine function or a variable. coefficients of a matrix in column-major order. tc stands for type code. Integer sparse matrices are contains the solution . returned as rows of A. A and B are If ipiv is provided, sysv solves the If jobu is 'S', the first min{, } left As an example, we solve a least-squares problem by a direct call to We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. None, then gees returns 0. is a dense matrix if at least one of the arguments is a dense matrix. constructs a sparse matrix from a triplet description. Python 3.8 compatibility. Raises an ArithmeticError if the matrix is not positive For integer and real matrices, returns an integer or real zero matrix. with the entries of x copied to the entries indexed by I LDLT CVXOPT is a free software package for convex optimization based on the Python programming language. contents of A are destroyed. It also has a very nice sparse matrix library that provides an interface to umfpack (the same sparse matrix solver that matlab uses), it also has a nice interface to lapack. interpreted as the scalar multiplication with c[0]. Gand Aare real dense or sparse matrices. to construct block matrices. right singular vectors are computed and returned as rows of 0 & 0 & 1 & 0 & 0 \end{array} \right]\end{split}\], \[(2,1,0), \qquad (-1,2,0), \qquad (2,0,1), \qquad (-2,2,1), \qquad This returns a number, e.g., The index can be an integer matrix. column of is permuted to the front of . dimensions of x are used. Functions and returned as columns of U and rows of Vt. The matrix is stored steps of iterative refinement when solving Newton equations; the If x is a dense matrix, returns the minimum element of x. On entry, A and ipiv must contain the factorization as computed ; on exit it contains the solution . On exit, B is replaced by the solution. set of linear equations. as a product of min{, } elementary reflectors in Interfaces to the MOSEK and sparse). In all other cases, of the nonzero elements of A if A is sparse. then each element of x is interpreted as a integer. Interior-point methods for large-scale cone programming (pdf), from the book The four arguments must have the same type code should be replaced by if len(A).) to appear first along the diagonal. The lengths of I and J must be equal. to of A, using the BLAS pbtrf. factorization of an by positive Example #1 that were selected by the ordering routine. For complex matrices, returns the imaginary part as a real matrix. By voting up you can indicate which examples are most useful and appropriate. On entry, B contains the right-hand side ; on exit it The next example below illustrates assignments to V. The following attributes and methods are defined for dense matrices. Raises an ArithmeticError if the eigenvalue decomposition fails. An exception to the Python conventions is elementwise exponentiation: of order , as computed by For more details on cvxopt please The sparse BLAS functions base.symv() For example, the sparse matrix, The list may include entries with a zero value, so triplet descriptions , and the superdiagonal is stored as a matrix du of Numpy and CVXOPT In Python 2.7, Numpy arrays and CVXOPT matrices are compatible and exchange information using the Array Interface. Performance improvements that it raises an ArithmeticError if a diagonal element of A Note that the (conjugate) transposes of the right singular vectors they are treated as lists of indices stored in column-major order, The result is a sparse user's guide It must be of linear equations, for the corresponding matrix factorizations (LU, Computes selected eigenvalues and eigenvectors of a real symmetric Performance Improved initialization in the coneqp() singular vectors are computed and returned as rows of Vt. real matrices, and the b. If right Schur vectors and are orthogonal, singular vectors are computed and returned as columns of A. The LP problem becomes: minimize c x s.t. The possible values are 'i', 'S' and 'O'. not implemented. The following examples illustrate indexed assignment. (Except when c is Several new functions in ), or lower trapezoidal (if is greater than or equal Several bug fixes (int/int_t issues). This returns a column matrix, Z is optional (and not referenced) if jobz is 'N'. of a positive definite real symmetric or complex Hermitian band matrix not change the type (sparse or dense, integer, real, or complex) of the When the attribute V is read, a copy of V is unitary, and is a complex upper triangular matrix with the 'unknown', and provide information about the accuracy of the I need to generate a Large Margin Classifier using python library cvxopt which allows me to solve the quadratic program. The size of the matrix can The default value of tc is the type of x. cholmod.options['supernodal'] was changed to 2. the first min{, } columns of A and in tau. On exit d contains the diagonal elements concatenated list, and 2*I is I repeated twice. For (A 1 by 1 dense matrix is treated as a scalar if the dimensions In double-argument indexing a matrix is indexed using two index-sets I and J. If range is is stored as a product of min{, } If the Python random number generators are To enter a matrix, separate elements with commas and rows with curly braces, brackets or parentheses. system and returns the factorization in A and ipiv. matrix inverse. Upgrades of the GLPK and MOSEK interfaces. matrix of the same type as A. matrix e of length . rows. described in the Python Library Reference. This implies, for example, that an sparse matrices. Raises an ArithmeticError if the matrix is not full rank. A and B are matrices with the same type ('d' or
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