Dense and Sparse Matrices
This chapter describes the two CVXOPT matrix types:
matrix
objects, used for dense matrix
computations, and spmatrix
objects, used for
sparse matrix computations.
Dense Matrices
A dense matrix is created by calling the function matrix
. The
arguments specify the values of the coefficients, the dimensions, and the
type (integer, double, or complex) of the matrix.
 cvxopt.matrix(x[, size[, tc]])
size
is a tuple of length two with the matrix dimensions. The number of rows and/or the number of columns can be zero.tc
stands for type code. The possible values are'i'
,'d'
, and'z'
, for integer, real (double), and complex matrices, respectively.x
can be a number, a sequence of numbers, a dense or sparse matrix, a one or twodimensional NumPy array, or a list of lists of matrices and numbers.If
x
is a number (Python integer, float, or complex number), a matrix is created with the dimensions specified bysize
and with all the coefficients equal tox
. The default value ofsize
is(1,1)
, and the default value oftc
is the type ofx
. If necessary, the type ofx
is converted (from integer to double when used to create a matrix of type'd'
, and from integer or double to complex when used to create a matrix of type'z'
).>>> from cvxopt import matrix >>> A = matrix(1, (1,4)) >>> print(A) [ 1 1 1 1] >>> A = matrix(1.0, (1,4)) >>> print(A) [ 1.00e+00 1.00e+00 1.00e+00 1.00e+00] >>> A = matrix(1+1j) >>> print(A) [ 1.00e+00+j1.00e+00]
If
x
is a sequence of numbers (list, tuple, array,array
array, …), then the numbers are interpreted as the coefficients of a matrix in columnmajor order. The length ofx
must be equal to the product ofsize[0]
andsize[1]
. Ifsize
is not specified, a matrix with one column is created. Iftc
is not specified, it is determined from the elements ofx
(and if that is impossible, for example becausex
is an empty list, a value'i'
is used). Type conversion takes place as for scalarx
.The following example shows several ways to define the same integer matrix.
>>> A = matrix([0, 1, 2, 3], (2,2)) >>> A = matrix((0, 1, 2, 3), (2,2)) >>> A = matrix(range(4), (2,2)) >>> from array import array >>> A = matrix(array('i', [0,1,2,3]), (2,2)) >>> print(A) [ 0 2] [ 1 3]
In Python 2.7 the following also works.
>>> A = matrix(xrange(4), (2,2))
If
x
is a dense or sparse matrix, then the coefficients ofx
are copied, in columnmajor order, to a new matrix of the given size. The total number of elements in the new matrix (the product ofsize[0]
andsize[1]
) must be the same as the product of the dimensions ofx
. Ifsize
is not specified, the dimensions ofx
are used. The default value oftc
is the type ofx
. Type conversion takes place when the type ofx
differs fromtc
, in a similar way as for scalarx
.>>> A = matrix([1., 2., 3., 4., 5., 6.], (2,3)) >>> print(A) [ 1.00e+00 3.00e+00 5.00e+00] [ 2.00e+00 4.00e+00 6.00e+00] >>> B = matrix(A, (3,2)) >>> print(B) [ 1.00e+00 4.00e+00] [ 2.00e+00 5.00e+00] [ 3.00e+00 6.00e+00] >>> C = matrix(B, tc='z') >>> print(C) [ 1.00e+00j0.00e+00 4.00e+00j0.00e+00] [ 2.00e+00j0.00e+00 5.00e+00j0.00e+00] [ 3.00e+00j0.00e+00 6.00e+00j0.00e+00]
NumPy arrays can be converted to matrices.
>>> from numpy import array >>> x = array([[1., 2., 3.], [4., 5., 6.]]) >>> x array([[ 1. 2. 3.] [ 4. 5. 6.]]) >>> print(matrix(x)) [ 1.00e+00 2.00e+00 3.00e+00] [ 4.00e+00 5.00e+00 6.00e+00]
If
x
is a list of lists of dense or sparse matrices and numbers (Python integer, float, or complex), then each element ofx
is interpreted as a blockcolumn stored in columnmajor order. Ifsize
is not specified, the blockcolumns are juxtaposed to obtain a matrix withlen(x)
blockcolumns. Ifsize
is specified, then the matrix withlen(x)
blockcolumns is resized by copying its elements in columnmajor order into a matrix of the dimensions given bysize
. Iftc
is not specified, it is determined from the elements ofx
(and if that is impossible, for example becausex
is a list of empty lists, a value'i'
is used). The same rules for type conversion apply as for scalarx
.>>> print(matrix([[1., 2.], [3., 4.], [5., 6.]])) [ 1.00e+00 3.00e+00 5.00e+00] [ 2.00e+00 4.00e+00 6.00e+00] >>> A1 = matrix([1, 2], (2,1)) >>> B1 = matrix([6, 7, 8, 9, 10, 11], (2,3)) >>> B2 = matrix([12, 13, 14, 15, 16, 17], (2,3)) >>> B3 = matrix([18, 19, 20], (1,3)) >>> C = matrix([[A1, 3.0, 4.0, 5.0], [B1, B2, B3]]) >>> print(C) [ 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]
A matrix with a single blockcolumn can be represented by a single list (i.e., if
x
is a list of lists, and has length one, then the argumentx
can be replaced byx[0]
).>>> D = matrix([B1, B2, B3]) >>> print(D) [ 6 8 10] [ 7 9 11] [ 12 14 16] [ 13 15 17] [ 18 19 20]
Sparse Matrices
A general spmatrix
object can be thought of as
a triplet description of a sparse matrix, i.e., a list of entries of the
matrix, with for each entry the value, row index, and column index.
Entries that are not included in the list are assumed to be zero.
For example, the sparse matrix
has the triplet description
The list may include entries with a zero value, so triplet descriptions are not necessarily unique. The list
is another triplet description of the same matrix.
An spmatrix
object corresponds to a particular
triplet description of a sparse matrix. We will refer to the entries in
the triplet description as the nonzero entries of the object,
even though they may have a numerical value zero.
Three functions are provided to create sparse matrices.
The first, spmatrix
,
constructs a sparse matrix from a triplet description.
 cvxopt.spmatrix(x, I, J[, size[, tc]])
I
andJ
are sequences of integers (lists, tuples,array
arrays, …) or integer matrices (matrix
objects with typecode'i'
), containing the row and column indices of the nonzero entries. The lengths ofI
andJ
must be equal. If they are matrices, they are treated as lists of indices stored in columnmajor order, i.e., as listslist(I)
, respectively,list(J)
.size
is a tuple of nonnegative integers with the row and column dimensions of the matrix. Thesize
argument is only needed when creating a matrix with a zero last row or last column. Ifsize
is not specified, it is determined fromI
andJ
: the default value forsize[0]
ismax(I)+1
ifI
is nonempty and zero otherwise. The default value forsize[1]
ismax(J)+1
ifJ
is nonempty and zero otherwise.tc
is the typecode,'d'
or'z'
, for double and complex matrices, respectively. Integer sparse matrices are not implemented.x
can be a number, a sequence of numbers, or a dense matrix. This argument specifies the numerical values of the nonzero entries.If
x
is a number (Python integer, float, or complex), a matrix is created with the sparsity pattern defined byI
andJ
, and nonzero entries initialized to the value ofx
. The default value oftc
is'd'
ifx
is integer or float, and'z'
ifx
is complex.The following code creates a 4 by 4 sparse identity matrix.
>>> from cvxopt import spmatrix >>> A = spmatrix(1.0, range(4), range(4)) >>> print(A) [ 1.00e+00 0 0 0 ] [ 0 1.00e+00 0 0 ] [ 0 0 1.00e+00 0 ] [ 0 0 0 1.00e+00]
If
x
is a sequence of numbers, a sparse matrix is created with the entries ofx
copied to the entries indexed byI
andJ
. The listx
must have the same length asI
andJ
. The default value oftc
is determined from the elements ofx
:'d'
ifx
contains integers and floatingpoint numbers or ifx
is an empty list, and'z'
ifx
contains at least one complex number.>>> A = spmatrix([2,1,2,2,1,4,3], [1,2,0,2,3,2,0], [0,0,1,1,2,3,4]) >>> print(A) [ 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 ]
If
x
is a dense matrix, a sparse matrix is created with all the entries ofx
copied, in columnmajor order, to the entries indexed byI
andJ
. The matrixx
must have the same length asI
andJ
. The default value oftc
is'd'
ifx
is an'i'
or'd'
matrix, and'z'
otherwise. IfI
andJ
contain repeated entries, the corresponding values of the coefficients are added.
The function sparse
constructs a sparse matrix
from a blockmatrix description.
 cvxopt.sparse(x[, tc])
tc
is the typecode,'d'
or'z'
, for double and complex matrices, respectively.x
can be amatrix
,spmatrix
, or a list of lists of matrices (matrix
orspmatrix
objects) and numbers (Python integer, float, or complex).If
x
is amatrix
orspmatrix
object, then a sparse matrix of the same size and the same numerical value is created. Numerical zeros inx
are treated as structural zeros and removed from the triplet description of the new sparse matrix.If
x
is a list of lists of matrices (matrix
orspmatrix
objects) and numbers (Python integer, float, or complex) then each element ofx
is interpreted as a (block)column matrix stored in colummajor order, and a blockmatrix is constructed by juxtaposing thelen(x)
blockcolumns (as inmatrix
). Numerical zeros are removed from the triplet description of the new matrix.
>>> from cvxopt import matrix, spmatrix, sparse >>> A = matrix([[1., 2., 0.], [2., 1., 2.], [0., 2., 1.]]) >>> print(A) [ 1.00e+00 2.00e+00 0.00e+00] [ 2.00e+00 1.00e+00 2.00e+00] [ 0.00e+00 2.00e+00 1.00e+00] >>> B = spmatrix([], [], [], (3,3)) >>> print(B) [0 0 0] [0 0 0] [0 0 0] >>> C = spmatrix([3, 4, 5], [0, 1, 2], [0, 1, 2]) >>> print(C) [ 3.00e+00 0 0 ] [ 0 4.00e+00 0 ] [ 0 0 5.00e+00] >>> D = sparse([[A, B], [B, C]]) >>> print(D) [ 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]
A matrix with a single blockcolumn can be represented by a single list.
>>> D = sparse([A, C]) >>> print(D) [ 1.00e+00 2.00e+00 0 ] [ 2.00e+00 1.00e+00 2.00e+00] [ 0 2.00e+00 1.00e+00] [ 3.00e+00 0 0 ] [ 0 4.00e+00 0 ] [ 0 0 5.00e+00]
The function spdiag
constructs a blockdiagonal
sparse matrix from a list of matrices.
 cvxopt.spdiag(x)
x
is a dense or sparse matrix with a single row or column, or a list of square dense or sparse matrices or scalars. Ifx
is a matrix, a sparse diagonal matrix is returned with the entries ofx
on its diagonal. Ifx
is list, a sparse blockdiagonal matrix is returned with the elements in the list as its diagonal blocks.>>> from cvxopt import matrix, spmatrix, spdiag >>> A = 3.0 >>> B = matrix([[1,2],[2,1]]) >>> C = spmatrix([1,1,1,1,1],[0,1,2,0,0,],[0,0,0,1,2]) >>> D = spdiag([A, B, C]) >>> print(D) [ 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 ] [ 0 0 0 1.00e+00 0 0 ]
Arithmetic Operations
The following table lists the arithmetic operations defined for dense and
sparse matrices. In the table A
and B
are dense or sparse
matrices of compatible dimensions, c
is a scalar (a Python number or
a dense 1 by 1 matrix), D
is a dense matrix, and e
is a Python
number.
Unary plus/minus 

Addition 

Subtraction 

Matrix multiplication 

Scalar multiplication and division 

Remainder after division 

Elementwise exponentiation 

The type of the result of these operations generally follows the Python
conventions.
For example, if A
and c
are integer, then in Python 2 the division
A/c
is interpreted as integer division and results in a
type 'i'
matrix, while in Python 3 it is interpreted as standard
divison and results in a type 'd'
matrix.
An exception to the Python conventions is elementwise exponentiation:
if D
is an integer matrix and e
is an integer number
than D**e
is a matrix of type 'd'
.
Addition, subtraction, and matrix multiplication with two matrix operands
result in a sparse matrix if both matrices are sparse, and in a dense
matrix otherwise. The result of a scalar multiplication or division is
dense if A
is dense, and sparse if A
is sparse. Postmultiplying
a matrix with a number c
means the same as premultiplying, i.e.,
scalar multiplication. Dividing a matrix by c
means dividing all its
entries by c
.
If c
in the expressions A+c
, c+A
, Ac
, cA
is a number,
then it is interpreted as a dense matrix with
the same dimensions as A
, type given by the type of c
, and all
its entries equal to c
. If c
is a 1 by 1 dense matrix and A
is not 1 by 1, then c
is interpreted as a dense matrix with the same
size of A
and all entries equal to c[0]
.
If c
is a 1 by 1 dense matrix, then, if possible, the products
c*A
and A*c
are interpreted as matrixmatrix products.
If the product cannot be interpreted as a matrixmatrix product
because the dimensions of A
are incompatible, then the product is
interpreted as the scalar multiplication with c[0]
.
The division A/c
and remainder A%c
with c
a
1 by 1 matrix are always interpreted as A/c[0]
, resp., A%c[0]
.
The following inplace operations are also defined, but only if they do
not change the type (sparse or dense, integer, real, or complex) of the
matrix A
. These inplace operations do not return a new matrix but
modify the existing object A
.
Inplace addition 

Inplace subtraction 

Inplace scalar multiplication and division 

Inplace remainder 

For example, if A
has typecode 'i'
, then A += B
is
allowed if B
has typecode 'i'
. It is not allowed if B
has typecode 'd'
or 'z'
because the addition
A+B
results in a 'd'
or 'z'
matrix and
therefore cannot be assigned to A
without changing its type.
As another example, if A
is a sparse matrix, then A += 1.0
is
not allowed because the operation A = A + 1.0
results in a dense
matrix, so it cannot be assigned to A
without changing its type.
Inplace matrixmatrix products are not allowed. (Except when c
is
a 1 by 1 dense matrix, in which case A *= c
is interpreted as the
scalar product A *= c[0]
.)
Inplace remainder is only defined for dense A
.
It is important to know when a matrix operation creates a new object. The following rules apply.
A simple assignment (
A = B
) is given the standard Python interpretation, i.e., it assigns to the variableA
a reference (or pointer) to the object referenced byB
.>>> B = matrix([[1.,2.], [3.,4.]]) >>> print(B) [ 1.00e+00 3.00e+00] [ 2.00e+00 4.00e+00] >>> A = B >>> A[0,0] = 1 >>> print(B) # modifying A[0,0] also modified B[0,0] [1.00e+00 3.00e+00] [ 2.00e+00 4.00e+00]
The regular (i.e., not inplace) arithmetic operations always return new objects.
>>> B = matrix([[1.,2.], [3.,4.]]) >>> A = +B >>> A[0,0] = 1 >>> print(B) # modifying A[0,0] does not modify B[0,0] [ 1.00e+00 3.00e+00] [ 2.00e+00 4.00e+00]
The inplace operations directly modify the coefficients of the existing matrix object and do not create a new object.
>>> B = matrix([[1.,2.], [3.,4.]]) >>> A = B >>> A *= 2 >>> print(B) # inplace operation also changed B [ 2.00e+00 6.00e+00] [ 4.00e+00 8.00e+00] >>> A = 2*A >>> print(B) # regular operation creates a new A, so does not change B [ 2.00e+00 6.00e+00] [ 4.00e+00 8.00e+00]
Indexing and Slicing
Matrices can be indexed using one or two arguments. In singleargument
indexing of a matrix A
, the index runs from len(A)
to
len(A)1
, and is interpreted as an index in the onedimensional
array of coefficients of A
in columnmajor order. Negative indices
have the standard Python interpretation: for negative k
,
A[k]
is the same element as A[len(A) + k]
.
Four different types of oneargument indexing are implemented.
The index can be a single integer. This returns a number, e.g.,
A[0]
is the first element ofA
.The index can be an integer matrix. This returns a column matrix: the command
A[matrix([0,1,2,3])]
returns the 4 by 1 matrix consisting of the first four elements ofA
. The size of the index matrix is ignored:A[matrix([0,1,2,3], (2,2))]
returns the same 4 by 1 matrix.The index can be a list of integers. This returns a column matrix, e.g.,
A[[0,1,2,3]]
is the 4 by 1 matrix consisting of elements 0, 1, 2, 3 ofA
.The index can be a Python slice. This returns a matrix with one column (possibly 0 by 1, or 1 by 1). For example,
A[::2]
is the column matrix defined by taking every other element ofA
, stored in columnmajor order.A[0:0]
is a matrix with size (0,1).
Thus, singleargument indexing returns a scalar (if the index is an integer), or a matrix with one column. This is consistent with the interpretation that singleargument indexing accesses the matrix in columnmajor order.
Note that an index list or an index matrix are equivalent, but they are
both useful, especially when we perform operations on index sets. For
example, if I
and J
are lists then I+J
is the
concatenated list, and 2*I
is I
repeated twice. If they
are matrices, these operations are interpreted as arithmetic operations.
For large index sets, indexing with integer matrices is also faster
than indexing with lists.
The following example illustrates oneargument indexing.
>>> from cvxopt import matrix, spmatrix
>>> A = matrix(range(16), (4,4), 'd')
>>> print(A)
[ 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]
>>> A[4]
4.0
>>> I = matrix([0, 5, 10, 15])
>>> print(A[I]) # the diagonal
[ 0.00e+00]
[ 5.00e+00]
[ 1.00e+01]
[ 1.50e+01]
>>> I = [0,2]; J = [1,3]
>>> print(A[2*I+J]) # duplicate I and append J
[ 0.00e+00]
[ 2.00e+00]
[ 0.00e+00]
[ 2.00e+00]
[ 1.00e+00]
[ 3.00e+00]
>>> I = matrix([0, 2]); J = matrix([1, 3])
>>> print(A[2*I+J]) # multiply I by 2 and add J
[ 1.00e+00]
[ 7.00e+00]
>>> print(A[4::4]) # get every fourth element skipping the first four
[ 4.00e+00]
[ 8.00e+00]
[ 1.20e+01]
In twoargument indexing the arguments can be any combinations of the four types listed above. The first argument indexes the rows of the matrix and the second argument indexes the columns. If both indices are scalars, then a scalar is returned. In all other cases, a matrix is returned. We continue the example.
>>> print(A[:,1])
[ 4.00e+00]
[ 5.00e+00]
[ 6.00e+00]
[ 7.00e+00]
>>> J = matrix([0, 2])
>>> print(A[J,J])
[ 0.00e+00 8.00e+00]
[ 2.00e+00 1.00e+01]
>>> print(A[:2, 2:])
[ 8.00e+00 1.20e+01]
[ 9.00e+00 1.30e+01]
>>> A = spmatrix([0,2,1,2,2,1], [0,1,2,0,2,1], [0,0,0,1,1,2])
>>> print(A[:, [0,1]])
[ 0.00e+00 2.00e+00]
[ 2.00e+00 0 ]
[1.00e+00 2.00e+00]
>>> B = spmatrix([0,2*1j,0,2], [1,2,1,2], [0,0,1,1,])
>>> print(B[2:,2:])
[ 0.00e+00j0.00e+00 0.00e+00j0.00e+00]
[ 0.00e+00+j2.00e+00 2.00e+00j0.00e+00]
Expressions of the form A[I]
or A[I,J]
can also appear on
the lefthand side of an assignment. The righthand side must be a
scalar
(i.e., a number or a 1 by 1 dense matrix), a sequence of numbers, or a
dense or sparse matrix. If the righthand side is a scalar, it is
interpreted as a dense matrix with identical entries and the dimensions of
the lefthand side. If the righthand side is a sequence of numbers
(list, tuple, array
array, range object, …) its values are
interpreted
as the coefficients of a dense matrix in columnmajor order. If the
righthand side is a matrix (matrix
or
spmatrix
), it must
have the same size as the lefthand side. Sparse matrices are
converted to dense in the assignment to a dense matrix.
Indexed assignments are only allowed if they do not change the type of
the matrix. For example, if A
is a matrix with type 'd'
,
then A[I] = B
is only permitted if B
is an integer, a float,
or a matrix of type 'i'
or 'd'
. If A
is an integer
matrix, then A[I] = B
is only permitted if B
is an integer or
an integer matrix.
The following examples illustrate indexed assignment.
>>> A = matrix(range(16), (4,4))
>>> A[::2,::2] = matrix([[1, 2], [3, 4]])
>>> print(A)
[ 1 4 3 12]
[ 1 5 9 13]
[ 2 6 4 14]
[ 3 7 11 15]
>>> A[::5] += 1
>>> print(A)
[ 0 4 3 12]
[ 1 6 9 13]
[ 2 6 3 14]
[ 3 7 11 16]
>>> A[0,:] = 1, 1, 1, 1
>>> print(A)
[ 1 1 1 1]
[ 1 6 9 13]
[ 2 6 3 14]
[ 3 7 11 16]
>>> A[2:,2:] = range(4)
>>> print(A)
[ 1 1 1 1]
[ 1 6 9 13]
[ 2 6 0 2]
[ 3 7 1 3]
>>> A = spmatrix([0,2,1,2,2,1], [0,1,2,0,2,1], [0,0,0,1,1,2])
>>> print(A)
[ 0.00e+00 2.00e+00 0 ]
[ 2.00e+00 0 1.00e+00]
[1.00e+00 2.00e+00 0 ]
>>> C = spmatrix([10,20,30], [0,2,1], [0,0,1])
>>> print(C)
[ 1.00e+01 0 ]
[ 0 3.00e+01]
[2.00e+01 0 ]
>>> A[:,0] = C[:,0]
>>> print(A)
[ 1.00e+01 2.00e+00 0 ]
[ 0 0 1.00e+00]
[2.00e+01 2.00e+00 0 ]
>>> D = matrix(range(6), (3,2))
>>> A[:,0] = D[:,0]
>>> print(A)
[ 0.00e+00 2.00e+00 0 ]
[ 1.00e+00 0 1.00e+00]
[ 2.00e+00 2.00e+00 0 ]
>>> A[:,0] = 1
>>> print(A)
[ 1.00e+00 2.00e+00 0 ]
[ 1.00e+00 0 1.00e+00]
[ 1.00e+00 2.00e+00 0 ]
>>> A[:,0] = 0
>>> print(A)
[ 0.00e+00 2.00e+00 0 ]
[ 0.00e+00 0 1.00e+00]
[ 0.00e+00 2.00e+00 0 ]
Attributes and Methods
Dense and sparse matrices have the following attributes.
 size
A tuple with the dimensions of the matrix. The size of the matrix can be changed by altering this attribute, as long as the number of elements in the matrix remains unchanged.
 typecode
A character, either
'i'
,'d'
, or'z'
, for integer, real, and complex matrices, respectively. A readonly attribute.
 trans()
Returns the transpose of the matrix as a new matrix. One can also use
A.T
instead ofA.trans()
.
 ctrans()
Returns the conjugate transpose of the matrix as a new matrix. One can also use
A.H
instead ofA.ctrans()
.
 real()
For complex matrices, returns the real part as a real matrix. For integer and real matrices, returns a copy of the matrix.
 imag()
For complex matrices, returns the imaginary part as a real matrix. For integer and real matrices, returns an integer or real zero matrix.
In addition, sparse matrices have the following attributes.
 V
A singlecolumn dense matrix containing the numerical values of the nonzero entries in columnmajor order. Making an assignment to the attribute is an efficient way of changing the values of the sparse matrix, without changing the sparsity pattern.
When the attribute
V
is read, a copy ofV
is returned, as a new dense matrix. This implies, for example, that an indexed assignmentA.V[I] = B
does not work, or at least cannot be used to modifyA
. Instead the attributeV
will be read and returned as a new matrix; then the elements of this new matrix are modified.
 I
A singlecolumn integer dense matrix with the row indices of the entries in
V
. A readonly attribute.
 J
A singlecolumn integer dense matrix with the column indices of the entries in
V
. A readonly attribute.
 CCS
A triplet (column pointers, row indices, values) with the compressedcolumnstorage representation of the matrix. A readonly attribute. This attribute can be used to export sparse matrices to other packages such as MOSEK.
The next example below illustrates assignments to V
.
>>> from cvxopt import spmatrix, matrix
>>> A = spmatrix(range(5), [0,1,1,2,2], [0,0,1,1,2])
>>> print(A)
[ 0.00e+00 0 0 ]
[ 1.00e+00 2.00e+00 0 ]
[ 0 3.00e+00 4.00e+00]
>>> B = spmatrix(A.V, A.J, A.I, (4,4)) # transpose and add a zero row and column
>>> print(B)
[ 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 ]
>>> B.V = matrix([1., 7., 8., 6., 4.]) # assign new values to nonzero entries
>>> print(B)
[ 1.00e+00 7.00e+00 0 0 ]
[ 0 8.00e+00 6.00e+00 0 ]
[ 0 0 4.00e+00 0 ]
[ 0 0 0 0 ]
The following attributes and methods are defined for dense matrices.
 tofile(f)
Writes the elements of the matrix in columnmajor order to a binary file
f
.
 fromfile(f)
Reads the contents of a binary file
f
into the matrix object.
The last two methods are illustrated in the following examples.
>>> from cvxopt import matrix, spmatrix
>>> A = matrix([[1.,2.,3.], [4.,5.,6.]])
>>> print(A)
[ 1.00e+00 4.00e+00]
[ 2.00e+00 5.00e+00]
[ 3.00e+00 6.00e+00]
>>> f = open('mat.bin','wb')
>>> A.tofile(f)
>>> f.close()
>>> B = matrix(0.0, (2,3))
>>> f = open('mat.bin','rb')
>>> B.fromfile(f)
>>> f.close()
>>> print(B)
[ 1.00e+00 3.00e+00 5.00e+00]
[ 2.00e+00 4.00e+00 6.00e+00]
>>> A = spmatrix(range(5), [0,1,1,2,2], [0,0,1,1,2])
>>> f = open('test.bin','wb')
>>> A.V.tofile(f)
>>> A.I.tofile(f)
>>> A.J.tofile(f)
>>> f.close()
>>> f = open('test.bin','rb')
>>> V = matrix(0.0, (5,1)); V.fromfile(f)
>>> I = matrix(0, (5,1)); I.fromfile(f)
>>> J = matrix(0, (5,1)); J.fromfile(f)
>>> B = spmatrix(V, I, J)
>>> print(B)
[ 0.00e+00 0 0 ]
[ 1.00e+00 2.00e+00 0 ]
[ 0 3.00e+00 4.00e+00]
Note that the dump
and load
functions in the pickle
module offer a convenient alternative for writing matrices to files and
reading matrices from files.
BuiltIn Functions
Many Python builtin functions and operations can be used with matrix arguments. We list some useful examples.
 len(x)
If
x
is a dense matrix, returns the product of the number of rows and the number of columns. Ifx
is a sparse matrix, returns the number of nonzero entries.
 bool([x])
Returns
False
ifx
is a zero matrix andTrue
otherwise.
 max(x)
If
x
is a dense matrix, returns the maximum element ofx
. Ifx
is a sparse, returns the maximum nonzero element ofx
.
 min(x)
If
x
is a dense matrix, returns the minimum element ofx
. Ifx
is a sparse matrix, returns the minimum nonzero element ofx
.
 abs(x)
Returns a matrix with the absolute values of the elements of
x
.
 sum(x[, start = 0.0])
Returns the sum of
start
and the elements ofx
.
Dense and sparse matrices can be used as arguments to the list
,
tuple
, zip
, map
, and filter
functions
described in the Python Library Reference.
However, one should note that when used with sparse matrix arguments,
these functions only consider the nonzero entries.
For example, list(A)
and tuple(A)
construct a list,
respectively a tuple, from the elements of A
if A
is dense, and
of the nonzero elements of A
if A
is sparse.
list(zip(A, B, ...))
returns a list of tuples, with the ith tuple
containing the ith elements (or nonzero elements) of A
, B
, ….
>>> from cvxopt import matrix
>>> A = matrix([[11., 5., 20.], [6., 0., 7.]])
>>> B = matrix(range(6), (3,2))
>>> list(A)
[11.0, 5.0, 20.0, 6.0, 0.0, 7.0]
>>> tuple(B)
(0, 1, 2, 3, 4, 5)
>>> list(zip(A, B))
[(11.0, 0), (5.0, 1), (20.0, 2), (6.0, 3), (0.0, 4), (7.0, 5)]
list(map(f, A))
, where f
is a function and A
is a
dense matrix, returns a list constructed by applying f
to each
element of A
. If
A
is sparse, the function f
is applied to each nonzero element of
A
. Multiple arguments can be provided, for example, as in
map(f, A, B)
, if f
is a function with two arguments.
In the following example, we return an integer 01 matrix with the
result of an elementwise comparison.
>>> A = matrix([ [0.5, 0.1, 2.0], [1.5, 0.2, 0.1], [0.3, 1.0, 0.0]])
>>> print(A)
[ 5.00e01 1.50e+00 3.00e01]
[1.00e01 2.00e01 1.00e+00]
[ 2.00e+00 1.00e01 0.00e+00]
>>> print(matrix(list(map(lambda x: 0 <= x <= 1, A)), A.size))
[ 1 0 1]
[ 0 1 1]
[ 0 0 1]
list(filter(f, A))
, where f
is a function and A
is a matrix,
returns a list containing the elements of A
(or nonzero elements of
A
is A
is sparse) for which f
is true.
>>> A = matrix([[5, 4, 10, 7], [1, 5, 6, 2], [6, 1, 5, 2], [1, 2, 3, 7]])
>>> print(A)
[ 5 1 6 1]
[ 4 5 1 2]
[ 10 6 5 3]
[ 7 2 2 7]
>>> list(filter(lambda x: x%2, A)) # list of odd elements in A
[5, 7, 1, 5, 1, 5, 1, 3, 7]
>>> list(filter(lambda x: 2 < x < 3, A)) # list of elements between 2 and 3
[1, 2, 1, 2, 1, 2]
It is also possible to iterate over matrix elements, as illustrated in the following example.
>>> A = matrix([[5, 3], [9, 11]])
>>> for x in A: print(max(x,0))
...
5
0
9
11
>>> [max(x,0) for x in A]
[5, 0, 9, 11]
The expression x in A
returns True
if an element
of A
(or a nonzero element of A
if A
is sparse)
is equal to x
and False
otherwise.
Other Matrix Functions
The following functions can be imported from CVXOPT.
 cvxopt.sqrt(x)
The elementwise square root of a dense matrix
x
. The result is returned as a real matrix ifx
is an integer or real matrix and as a complex matrix ifx
is a complex matrix. Raises an exception whenx
is an integer or real matrix with negative elements.As an example we take the elementwise square root of the sparse matrix
>>> from cvxopt import spmatrix, sqrt >>> A = spmatrix([2,1,2,2,1,3,4], [1,2,0,2,3,0,2], [0,0,1,1,2,3,3]) >>> B = spmatrix(sqrt(A.V), A.I, A.J) >>> print(B) [ 0 1.41e+00 0 1.73e+00] [ 1.41e+00 0 0 0 ] [ 1.00e+00 1.41e+00 0 2.00e+00] [ 0 0 1.00e+00 0 ]
 cvxopt.sin(x)
The sine function applied elementwise to a dense matrix
x
. The result is returned as a real matrix ifx
is an integer or real matrix and as a complex matrix otherwise.
 cvxopt.cos(x)
The cosine function applied elementwise to a dense matrix
x
. The result is returned as a real matrix ifx
is an integer or real matrix and as a complex matrix otherwise.
 cvxopt.exp(x)
The exponential function applied elementwise to a dense matrix
x
. The result is returned as a real matrix ifx
is an integer or real matrix and as a complex matrix otherwise.
 cvxopt.log(x)
The natural logarithm applied elementwise to a dense matrix
x
. The result is returned as a real matrix ifx
is an integer or real matrix and as a complex matrix otherwise. Raises an exception whenx
is an integer or real matrix with nonpositive elements, or a complex matrix with zero elements.
 cvxopt.mul(x0[, x1[, x2 ...]])
If the arguments are dense or sparse matrices of the same size, returns the elementwise product of its arguments. The result is a sparse matrix if one or more of its arguments is sparse, and a dense matrix otherwise.
If the arguments include scalars, a scalar product with the scalar is made. (A 1 by 1 dense matrix is treated as a scalar if the dimensions of the other arguments are not all 1 by 1.)
mul
can also be called with an iterable (list, tuple, range object, or generator) as its single argument, if the iterable generates a list of dense or sparse matrices or scalars.>>> from cvxopt import matrix, spmatrix, mul >>> A = matrix([[1.0, 2.0], [3.0, 4.0]]) >>> B = spmatrix([2.0, 3.0], [0, 1], [0, 1]) >>> print(mul(A, B, 1.0)) [2.00e+00 0 ] [ 0 1.20e+01] >>> print(mul( matrix([k, k+1]) for k in [1,2,3] )) [ 6] [ 24]
 cvxopt.div(x, y)
Returns the elementwise division of
x
byy
.x
is a dense or sparse matrix, or a scalar (Python number of 1 by 1 dense matrix).y
is a dense matrix or a scalar.
 cvxopt.max(x0[, x1[, x2 ...]])
When called with a single matrix argument, returns the maximum of the elements of the matrix (including the zero entries, if the matrix is sparse).
When called with multiple arguments, the arguments must be matrices of the same size, or scalars, and the elementwise maximum is returned. A 1 by 1 dense matrix is treated as a scalar if the other arguments are not all 1 by 1. If one of the arguments is scalar, and the other arguments are not all 1 by 1, then the scalar argument is interpreted as a dense matrix with all its entries equal to the scalar.
The result is a sparse matrix if all its arguments are sparse matrices. The result is a number if all its arguments are numbers. The result is a dense matrix if at least one of the arguments is a dense matrix.
max
can also be called with an iterable (list, tuple, range object, or generator) as its single argument, if the iterable generates a list of dense or sparse matrices or scalars.>>> from cvxopt import matrix, spmatrix, max >>> A = spmatrix([2, 3], [0, 1], [0, 1]) >>> print(max(A, A, 1)) [ 2.00e+00 1.00e+00] [ 1.00e+00 3.00e+00]
It is important to note the difference between this
max
and the builtinmax
, explained in the previous section.>>> from cvxopt import spmatrix >>> A = spmatrix([1.0, 2.0], [0,1], [0,1]) >>> max(A) # builtin max of a sparse matrix takes maximum over nonzero elements 1.0 >>> max(A, 1.5) Traceback (most recent call last): File "<stdin>", line 1, in <module> NotImplementedError: matrix comparison not implemented >>> from cvxopt import max >>> max(A) # cvxopt.max takes maximum over all the elements 0.0 >>> print(max(A, 1.5)) [1.00e+00 0.00e+00] [ 0.00e+00 1.50e+00]
 cvxopt.min(x0[, x1[, x2 ...]])
When called with a single matrix argument, returns the minimum of the elements of the matrix (including the zero entries, if the matrix is sparse).
When called with multiple arguments, the arguments must be matrices of the same size, or scalars, and the elementwise maximum is returned. A 1 by 1 dense matrix is treated as a scalar if the other arguments are not all 1 by 1. If one of the arguments is scalar, and the other arguments are not all 1 by 1, then the scalar argument is interpreted as a dense matrix with all its entries equal to the scalar.
min
can also be called with an iterable (list, tuple, range object, or generator) as its single argument, if the iterable generates a list of dense or sparse matrices or scalars.
Randomly Generated Matrices
The CVXOPT package provides two functions
normal
and
uniform
for generating randomly distributed
matrices.
The default installation relies on the pseudorandom number generators in
the Python standard library random
. Alternatively, the random
number generators in the
GNU Scientific Library (GSL)
can be used, if this option is selected during the installation of CVXOPT.
The random matrix functions based on GSL are faster than the default
functions based on the random
module.
 cvxopt.normal(nrows[, ncols = 1[, mean = 0.0[, std = 1.0]]])
Returns a type
'd'
dense matrix of sizenrows
byncols
with elements chosen from a normal distribution with meanmean
and standard deviationstd
.
 cvxopt.uniform(nrows[, ncols = 1[, a = 0.0[, b = 1.0]]])
Returns a type
'd'
dense matrix of sizenrows
byncols
matrix with elements uniformly distributed betweena
andb
.
 cvxopt.setseed([value])
Sets the state of the random number generator.
value
must be an integer. Ifvalue
is absent or equal to zero, the value is taken from the system clock. If the Python random number generators are used, this is equivalent torandom.seed(value)
.
 cvxopt.getseed()
Returns the current state of the random number generator. This function is only available if the GSL random number generators are installed. (The state of the random number generators in the Python
random
module can be managed via the functionsrandom.getstate
andrandom.setstate
.)