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Compute the upper Cholesky factor, R, of the real symmetric or complex Hermitian positive definite matrix A.
The upper Cholesky factor R is computed by using the upper triangular part of matrix A and is defined by
R' * R = A.
Calling chol
using the optional "upper"
flag has the
same behavior. In contrast, using the optional "lower"
flag,
chol
returns the lower triangular factorization, computed by using
the lower triangular part of matrix A, such that
L * L' = A.
Called with one output argument chol
fails if matrix A is
not positive definite. Note that if matrix A is not real symmetric
or complex Hermitian then the lower triangular part is considered to be
the (complex conjugate) transpose of the upper triangular part, or vice
versa, given the "lower"
flag.
Called with two or more output arguments p flags whether the matrix
A was positive definite and chol
does not fail. A zero value
of p indicates that matrix A is positive definite and R
gives the factorization. Otherwise, p will have a positive value.
If called with three output arguments matrix A must be sparse and
a sparsity preserving row/column permutation is applied to matrix A
prior to the factorization. That is R is the factorization of
A(Q,Q)
such that
R' * R = Q' * A * Q.
The sparsity preserving permutation is generally returned as a matrix.
However, given the optional flag "vector"
, Q will be
returned as a vector such that
R' * R = A(Q, Q).
In general the lower triangular factorization is significantly faster for sparse matrices.
See also: hess, lu, qr, qz, schur, svd, ichol, cholinv, chol2inv, cholupdate, cholinsert, choldelete, cholshift.
Compute the inverse of the symmetric positive definite matrix A using the Cholesky factorization.
Invert a symmetric, positive definite square matrix from its Cholesky decomposition, U.
Note that U should be an upper-triangular matrix with positive
diagonal elements. chol2inv (U)
provides
inv (U'*U)
but it is much faster than using inv
.
Update or downdate a Cholesky factorization.
Given an upper triangular matrix R and a column vector u, attempt to determine another upper triangular matrix R1 such that
"+"
"-"
If op is "-"
, info is set to
If info is not present, an error message is printed in cases 1 and 2.
See also: chol, cholinsert, choldelete, cholshift.
Update a Cholesky factorization given a row or column to insert in the original factored matrix.
Given a Cholesky factorization of a real symmetric or complex Hermitian positive definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of A1, where A1(p,p) = A, A1(:,j) = A1(j,:)’ = u and p = [1:j-1,j+1:n+1]. u(j) should be positive.
On return, info is set to
If info is not present, an error message is printed in cases 1 and 2.
See also: chol, cholupdate, choldelete, cholshift.
Update a Cholesky factorization given a row or column to delete from the original factored matrix.
Given a Cholesky factorization of a real symmetric or complex Hermitian positive definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of A(p,p), where p = [1:j-1,j+1:n+1].
See also: chol, cholupdate, cholinsert, cholshift.
Update a Cholesky factorization given a range of columns to shift in the original factored matrix.
Given a Cholesky factorization of a real symmetric or complex Hermitian
positive definite matrix A = R’*R, R upper
triangular, return the Cholesky factorization of
A(p,p), where p is the permutation
p = [1:i-1, shift(i:j, 1), j+1:n]
if i < j
or
p = [1:j-1, shift(j:i,-1), i+1:n]
if j < i.
See also: chol, cholupdate, cholinsert, choldelete.
Compute the Hessenberg decomposition of the matrix A.
The Hessenberg decomposition is
P * H * P' = A
where P is a square
unitary matrix (P' * P = I
, using complex-conjugate
transposition) and H is upper Hessenberg
(H(i, j) = 0 forall i > j+1)
.
The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979).
Compute the LU decomposition of A.
If A is full then subroutines from LAPACK are used, and if A is sparse then UMFPACK is used.
The result is returned in a permuted form, according to the optional return
value P. For example, given the matrix A = [1, 2; 3, 4]
,
[L, U, P] = lu (A)
returns
L = 1.00000 0.00000 0.33333 1.00000 U = 3.00000 4.00000 0.00000 0.66667 P = 0 1 1 0
The matrix is not required to be square.
When called with two or three output arguments and a sparse input matrix,
lu
does not attempt to perform sparsity preserving column permutations.
Called with a fourth output argument, the sparsity preserving column
transformation Q is returned, such that
P * A * Q = L * U
. This is the
preferred way to call lu
with sparse input matrices.
Called with a fifth output argument and a sparse input matrix, lu
attempts to use a scaling factor R on the input matrix such that
P * (R \ A) * Q = L * U
.
This typically leads to a sparser and more stable factorization.
An additional input argument thresh that defines the pivoting
threshold can be given. thresh can be a scalar, in which case
it defines the UMFPACK pivoting tolerance for both symmetric and
unsymmetric cases. If thresh is a 2-element vector, then the first
element defines the pivoting tolerance for the unsymmetric UMFPACK
pivoting strategy and the second for the symmetric strategy. By default,
the values defined by spparms
are used ([0.1, 0.001]).
Given the string argument "vector"
, lu
returns the values
of P and Q as vector values, such that for full matrix,
A(P,:) = L * U
, and R(P,:)
* A(:,Q) = L * U
.
With two output arguments, returns the permuted forms of the upper and
lower triangular matrices, such that A = L * U
.
With one output argument y, then the matrix returned by the
LAPACK routines is returned. If the input matrix is sparse then the
matrix L is embedded into U to give a return value similar to
the full case. For both full and sparse matrices, lu
loses the
permutation information.
Given an LU factorization of a real or complex matrix A = L*U, L lower unit trapezoidal and U upper trapezoidal, return the LU factorization of A + x*y.’, where x and y are column vectors (rank-1 update) or matrices with equal number of columns (rank-k update).
Optionally, row-pivoted updating can be used by supplying a row permutation
(pivoting) matrix P; in that case, an updated permutation matrix is
returned. Note that if L, U, P is a pivoted
LU factorization as obtained by lu
:
[L, U, P] = lu (A);
then a factorization of A+x*y.'
can be obtained
either as
[L1, U1] = lu (L, U, P*x, y)
or
[L1, U1, P1] = lu (L, U, P, x, y)
The first form uses the unpivoted algorithm, which is faster, but less stable. The second form uses a slower pivoted algorithm, which is more stable.
The matrix case is done as a sequence of rank-1 updates; thus, for large enough k, it will be both faster and more accurate to recompute the factorization from scratch.
See also: lu, cholupdate, qrupdate.
Compute the QR factorization of A, using standard LAPACK subroutines.
The QR factorization is
Q * R = A
where Q is an orthogonal matrix and R is upper triangular.
For example, given the matrix A = [1, 2; 3, 4]
,
[Q, R] = qr (A)
returns
Q = -0.31623 -0.94868 -0.94868 0.31623 R = -3.16228 -4.42719 0.00000 -0.63246
which multiplied together return the original matrix
Q * R ⇒ 1.0000 2.0000 3.0000 4.0000
If just a single return value is requested then it is either R, if
A is sparse, or X, such that R = triu (X)
if
A is full. (Note: unlike most commands, the single return value is not
the first return value when multiple values are requested.)
If a third output P is requested, then qr
calculates the permuted
QR factorization
Q * R = A * P
where Q is an orthogonal matrix, R is upper triangular, and P is a permutation matrix.
If A is dense, the permuted QR factorization has the additional
property that the diagonal entries of R are ordered by decreasing
magnitude. In other words, abs (diag (R))
will be ordered
from largest to smallest.
If A is sparse, P is a fill-reducing ordering of the columns of A. In that case, the diagonal entries of R are not ordered by decreasing magnitude.
For example, given the matrix A = [1, 2; 3, 4]
,
[Q, R, P] = qr (A)
returns
Q = -0.44721 -0.89443 -0.89443 0.44721 R = -4.47214 -3.13050 0.00000 0.44721 P = 0 1 1 0
If the input matrix A is sparse, the sparse QR factorization
is computed by using SPQR or CXSPARSE (e.g., if SPQR is not
available). Because the matrix Q is, in general, a full matrix, it is
recommended to request only one return value R. In that case, the
computation avoids the construction of Q and returns a sparse R
such that R = chol (A' * A)
.
If A is dense, an additional matrix B is supplied and two
return values are requested, then qr
returns C, where
C = Q' * B
. This allows the least squares
approximation of A \ B
to be calculated as
[C, R] = qr (A, B) X = R \ C
If A is a sparse MxN matrix and an additional matrix B is
supplied, one or two return values are possible. If one return value X
is requested and M < N, then X is the minimum 2-norm solution of
A \ B
. If M >= N, X is the least squares
approximation of A \ B
. If two return values are
requested, C and R have the same meaning as in the dense case
(C is dense and R is sparse).
The version with one return parameter should be preferred because
it uses less memory and can handle rank-deficient matrices better.
If the final argument is the string "vector"
then P is a
permutation vector (of the columns of A) instead of a permutation
matrix. In this case, the defining relationship is:
Q * R = A(:, P)
The default, however, is to return a permutation matrix and this may be
explicitly specified by using a final argument of "matrix"
.
If the final argument is the scalar 0 an "economy"
factorization is
returned. If the original matrix A has size MxN and M > N, then the
"economy"
factorization will calculate just N rows in R and N
columns in Q and omit the zeros in R. If M ≤ N, there is no
difference between the economy and standard factorizations. When calculating
an "economy"
factorization and A is dense, the output P is
always a vector rather than a matrix. If A is sparse, output
P is a sparse permutation matrix.
Background: The QR factorization has applications in the solution of least squares problems
min norm (A*x - b)
for overdetermined systems of equations (i.e., A is a tall, thin matrix).
The permuted QR factorization
[Q, R, P] = qr (A)
allows the construction of an
orthogonal basis of span (A)
.
See also: chol, hess, lu, qz, schur, svd, qrupdate, qrinsert, qrdelete, qrshift.
Update a QR factorization given update vectors or matrices.
Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and R upper trapezoidal, return the QR factorization of A + u*v’, where u and v are column vectors (rank-1 update) or matrices with equal number of columns (rank-k update). Notice that the latter case is done as a sequence of rank-1 updates; thus, for k large enough, it will be both faster and more accurate to recompute the factorization from scratch.
The QR factorization supplied may be either full (Q is square) or economized (R is square).
Update a QR factorization given a row or column to insert in the original factored matrix.
Given a QR factorization of a real or complex matrix
A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of
[A(:,1:j-1) x A(:,j:n)], where u is a column vector to be inserted
into A (if orient is "col"
), or the
QR factorization of [A(1:j-1,:);x;A(:,j:n)], where x is a row
vector to be inserted into A (if orient is "row"
).
The default value of orient is "col"
. If orient is
"col"
, u may be a matrix and j an index vector
resulting in the QR factorization of a matrix B such that
B(:,j) gives u and B(:,j) = [] gives A.
Notice that the latter case is done as a sequence of k insertions;
thus, for k large enough, it will be both faster and more accurate to
recompute the factorization from scratch.
If orient is "col"
, the QR factorization supplied may
be either full (Q is square) or economized (R is square).
If orient is "row"
, full factorization is needed.
Update a QR factorization given a row or column to delete from the original factored matrix.
Given a QR factorization of a real or complex matrix
A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of
[A(:,1:j-1), U, A(:,j:n)],
where u is a column vector to be inserted into A
(if orient is "col"
),
or the QR factorization of [A(1:j-1,:);X;A(:,j:n)],
where x is a row orient is "row"
).
The default value of orient is "col"
.
If orient is "col"
, j may be an index vector
resulting in the QR factorization of a matrix B such that
A(:,j) = [] gives B. Notice that the latter case is done as
a sequence of k deletions; thus, for k large enough, it will be both faster
and more accurate to recompute the factorization from scratch.
If orient is "col"
, the QR factorization supplied may
be either full (Q is square) or economized (R is square).
If orient is "row"
, full factorization is needed.
Update a QR factorization given a range of columns to shift in the original factored matrix.
Given a QR factorization of a real or complex matrix
A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization
of A(:,p), where p is the permutation
p = [1:i-1, shift(i:j, 1), j+1:n]
if i < j
or
p = [1:j-1, shift(j:i,-1), i+1:n]
if j < i.
Compute the QZ decomposition of a generalized eigenvalue problem.
The generalized eigenvalue problem is defined as
A x = lambda B x
There are three calling forms of the function:
lambda = qz (A, B)
Compute the generalized eigenvalues lambda.
[AA, BB, Q, Z, V, W, lambda] = qz (A, B)
Compute QZ decomposition, generalized eigenvectors, and generalized eigenvalues.
AA = Q * A * Z, BB = Q * B * Z A * V = B * V * diag (lambda) W' * A = diag (lambda) * W' * B
with Q and Z orthogonal (unitary for complex case).
[AA, BB, Z {, lambda}] = qz (A, B, opt)
As in form 2 above, but allows ordering of generalized eigenpairs for, e.g., solution of discrete time algebraic Riccati equations. Form 3 is not available for complex matrices, and does not compute the generalized eigenvectors V, W, nor the orthogonal matrix Q.
for ordering eigenvalues of the GEP pencil. The leading block of the revised pencil contains all eigenvalues that satisfy:
"N"
unordered (default)
"S"
small: leading block has all |lambda| < 1
"B"
big: leading block has all |lambda| ≥ 1
"-"
negative real part: leading block has all eigenvalues in the open left half-plane
"+"
non-negative real part: leading block has all eigenvalues in the closed right half-plane
Note: qz
performs permutation balancing, but not scaling
(see balance
), which may be lead to less accurate
results than eig
. The order of output arguments was selected for
compatibility with MATLAB.
See also: eig, gsvd, balance, chol, hess, lu, qr, qzhess, schur.
Compute the Hessenberg-triangular decomposition of the matrix pencil
(A, B)
, returning
aa = q * A * z
,
bb = q * B * z
, with q and z
orthogonal.
For example:
[aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8]) ⇒ aa = -3.02244 -4.41741 0.92998 0.69749 ⇒ bb = -8.60233 -9.99730 0.00000 -0.23250 ⇒ q = -0.58124 -0.81373 -0.81373 0.58124 ⇒ z = Diagonal Matrix 1 0 0 1
The Hessenberg-triangular decomposition is the first step in Moler and Stewart’s QZ decomposition algorithm.
Algorithm taken from Golub and Van Loan, Matrix Computations, 2nd edition.
Compute the Schur decomposition of A.
The Schur decomposition is defined as
S = U' * A * U
where U is a unitary matrix
(U'* U
is identity)
and S is upper triangular. The eigenvalues of A (and S)
are the diagonal elements of S. If the matrix A is real, then
the real Schur decomposition is computed, in which the matrix U
is orthogonal and S is block upper triangular with blocks of size at
most
2 x 2
along the diagonal. The diagonal elements of S
(or the eigenvalues of the
2 x 2
blocks, when appropriate) are the eigenvalues of A and S.
The default for real matrices is a real Schur decomposition.
A complex decomposition may be forced by passing the flag
"complex"
.
The eigenvalues are optionally ordered along the diagonal according to the
value of opt. opt = "a"
indicates that all eigenvalues
with negative real parts should be moved to the leading block of S
(used in are
), opt = "d"
indicates that all
eigenvalues with magnitude less than one should be moved to the leading
block of S (used in dare
), and opt = "u"
, the
default, indicates that no ordering of eigenvalues should occur. The
leading k columns of U always span the A-invariant
subspace corresponding to the k leading eigenvalues of S.
The Schur decomposition is used to compute eigenvalues of a square
matrix, and has applications in the solution of algebraic Riccati
equations in control (see are
and dare
).
See also: rsf2csf, ordschur, ordeig, lu, chol, hess, qr, qz, svd.
Convert a real, upper quasi-triangular Schur form TR to a complex, upper triangular Schur form T.
Note that the following relations hold:
UR * TR * UR' = U * T * U'
and
U' * U
is the identity matrix I.
Note also that U and T are not unique.
See also: schur.
Reorders the real Schur factorization (U,S) obtained with the
schur
function, so that selected eigenvalues appear in the upper left
diagonal blocks of the quasi triangular Schur matrix.
The logical vector select specifies the selected eigenvalues as they appear along S’s diagonal.
For example, given the matrix A = [1, 2; 3, 4]
, and its Schur
decomposition
[U, S] = schur (A)
which returns
U = -0.82456 -0.56577 0.56577 -0.82456 S = -0.37228 -1.00000 0.00000 5.37228
It is possible to reorder the decomposition so that the positive eigenvalue is in the upper left corner, by doing:
[U, S] = ordschur (U, S, [0,1])
Reorder the QZ decomposition of a generalized eigenvalue problem.
The generalized eigenvalue problem is defined as
A x = lambda B x
Its generalized Schur decomposition is computed using the qz
algorithm:
[AA, BB, Q, Z] = qz (A, B)
where AA, BB, Q, and Z fulfill
AA = Q * A * Z, BB = Q * B * Z
The ordqz
function computes a unitary transformation QR and
ZR such that the order of the eigenvalue on the diagonal of AA and
BB is changed. The resulting reordered matrices AR and BR
fulfill:
AR = QR * A * ZR, BR = QR * B * ZR
The function can either be called with the keyword argument which selects the eigenvalues in the top left block of AR and BR in the following way:
"S"
, "udi"
small: leading block has all |lambda| < 1
"B"
, "udo"
big: leading block has all |lambda| ≥ 1
"-"
, "lhp"
negative real part: leading block has all eigenvalues in the open left half-plane
"+"
, "rhp"
non-negative real part: leading block has all eigenvalues in the closed right half-plane
If a logical vector select is given instead of a keyword the ordqz
function reorders all eigenvalues k
to the left block for which
select(k)
is true.
Note: The keywords are compatible with the ones from qr
.
Return the eigenvalues of quasi-triangular matrices in their order of appearance in the matrix A.
The quasi-triangular matrix A is usually the result of a Schur
factorization. If called with a second input B then the generalized
eigenvalues of the pair A, B are returned in the order of
appearance of the matrix A-lambda*B
. The pair
A, B is usually the result of a QZ decomposition.
Determine the largest principal angle between two subspaces spanned by the columns of matrices A and B.
Compute the singular value decomposition of A.
The singular value decomposition is defined by the relation
A = U*S*V'
The function svd
normally returns only the vector of singular values.
When called with three return values, it computes
U, S, and V.
For example,
svd (hilb (3))
returns
ans = 1.4083189 0.1223271 0.0026873
and
[u, s, v] = svd (hilb (3))
returns
u = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867 s = 1.40832 0.00000 0.00000 0.00000 0.12233 0.00000 0.00000 0.00000 0.00269 v = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867
When given a second argument that is not 0, svd
returns an economy-sized
decomposition, eliminating the unnecessary rows or columns of U or
V.
If the second argument is exactly 0, then the choice of decomposition is based on the matrix A. If A has more rows than columns then an economy-sized decomposition is returned, otherwise a regular decomposition is calculated.
Algorithm Notes: When calculating the full decomposition (left and right
singular matrices in addition to singular values) there is a choice of two
routines in LAPACK. The default routine used by Octave is gesvd
.
The alternative is gesdd
which is 5X faster, but may use more memory
and may be inaccurate for some input matrices. There is a third routine
gejsv
, suitable for better accuracy at extreme scale. See the
documentation for svd_driver
for more information on choosing a driver.
Query or set the underlying LAPACK driver used by svd
.
Currently recognized values are "gesdd"
, "gesvd"
, and
"gejsv"
. The default is "gesvd"
.
When called from inside a function with the "local"
option, the
variable is changed locally for the function and any subroutines it calls.
The original variable value is restored when exiting the function.
Algorithm Notes: The LAPACK library routines gesvd
and gesdd
are different only when calculating the full singular value decomposition (left
and right singular matrices as well as singular values). When calculating just
the singular values the following discussion is not relevant.
The newer gesdd
routine is based on a Divide-and-Conquer algorithm that
is 5X faster than the alternative gesvd
, which is based on QR
factorization. However, the new algorithm can use significantly more memory.
For an MxN input matrix the memory usage is of order O(min(M,N) ^ 2),
whereas the alternative is of order O(max(M,N)).
The routine gejsv
uses a preconditioned Jacobi SVD algorithm. Unlike
gesvd
and gesdd
, in gejsv
, there is no bidiagonalization
step that could contaminate accuracy in some extreme cases. Also, gejsv
is known to be optimally accurate in some sense. However, the speed is slower
(single threaded at its core) and uses more memory (O(min(M,N) ^ 2 + M + N)).
Beyond speed and memory issues, there have been instances where some input
matrices were not accurately decomposed by gesdd
. See currently active
bug https://savannah.gnu.org/bugs/?55564. Until these accuracy issues
are resolved in a new version of the LAPACK library, the default driver
in Octave has been set to "gesvd"
.
See also: svd.
Compute Householder reflection vector housv to reflect x to be the j-th column of identity, i.e.,
(I - beta*housv*housv')x = norm (x)*e(j) if x(j) < 0, (I - beta*housv*housv')x = -norm (x)*e(j) if x(j) >= 0
Inputs
vector
index into vector
threshold for zero (usually should be the number 0)
Outputs (see Golub and Van Loan):
If beta = 0, then no reflection need be applied (zer set to 0)
householder vector
Construct an orthogonal basis u of a block Krylov subspace.
The block Krylov subspace has the following form:
[v a*v a^2*v … a^(k+1)*v]
The construction is made with Householder reflections to guard against loss of orthogonality.
If V is a vector, then h contains the Hessenberg matrix
such that a*u == u*h+rk*ek'
, in which
rk = a*u(:,k)-u*h(:,k)
, and ek'
is the vector
[0, 0, …, 1]
of length k. Otherwise, h is
meaningless.
If V is a vector and k is greater than length (A) - 1
,
then h contains the Hessenberg matrix such that a*u == u*h
.
The value of nu is the dimension of the span of the Krylov subspace (based on eps1).
If b is a vector and k is greater than m-1, then h contains the Hessenberg decomposition of A.
The optional parameter eps1 is the threshold for zero. The default value is 1e-12.
If the optional parameter pflg is nonzero, row pivoting is used to improve numerical behavior. The default value is 0.
Reference: A. Hodel, P. Misra, Partial Pivoting in the Computation of Krylov Subspaces of Large Sparse Systems, Proceedings of the 42nd IEEE Conference on Decision and Control, December 2003.
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