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### 18.3 Matrix Factorizations

— Loadable Function: chol (a)

> Compute the Cholesky factor, r, of the symmetric positive definite matrix a, where

>               r' * r = a.

> See also: cholinv, chol2inv.

— Loadable Function: cholinv (a)

> Use the Cholesky factorization to compute the inverse of the symmetric positive definite matrix a.

> See also: chol, chol2inv.

— Loadable Function: chol2inv (u)

> 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`.

> See also: chol, cholinv.

— Loadable Function: h = hess (a)
— Loadable Function: [p, h] = hess (a)

> Compute the Hessenberg decomposition of the matrix a.

> 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). 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 (`i >= j+1 => h (i, j) = 0`).

— Loadable Function: [l, u, p] = lu (a)

> Compute the LU decomposition of a, using subroutines from Lapack. 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.

— Loadable Function: [q, r, p] = qr (a)

> Compute the QR factorization of a, using standard Lapack subroutines. 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

> 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 QR factorization is `q * r = a` where `q` is an orthogonal matrix and `r` is upper triangular.

> The permuted QR factorization `[`q`,` r`,` p`] = qr (`a`)` forms the QR factorization such that the diagonal entries of `r` are decreasing in magnitude order. 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

> The permuted `qr` factorization `[q, r, p] = qr (a)` factorization allows the construction of an orthogonal basis of `span (a)`.

— Loadable Function: lambda = qz (a, b)

> Generalized eigenvalue problem A x = s B x, QZ decomposition. There are three ways to call this function:

> 1. `lambda = qz(A,B)`

>    Computes the generalized eigenvalues lambda of (A - s B).

> 2. `[AA, BB, Q, Z, V, W, lambda] = qz (A, B)`

>    Computes qz decomposition, generalized eigenvectors, and generalized eigenvalues of (A - sB)

>                           A*V = B*V*diag(lambda)
>                           W'*A = diag(lambda)*W'*B
>                           AA = Q'*A*Z, BB = Q'*B*Z

>    with Q and Z orthogonal (unitary)= I

> 3. `[AA,BB,Z{, lambda}] = qz(A,B,opt)`

>    As in form [2], 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.

>    opt
>      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
>      `"+"`
>        = nonnegative real part: leading block has all eigenvalues in the closed right half-plane

> Note: qz performs permutation balancing, but not scaling (see balance). Order of output arguments was selected for compatibility with MATLAB

> See also: balance, dare, eig, schur.

— Function File: [aa, bb, q, z] = qzhess (a, b)

> 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 = [ 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.

— Loadable Function: s = schur (a)
— Loadable Function: [u, s] = schur (a, opt)

> 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`). `schur` always returns `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 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`.

— Loadable Function: s = svd (a)
— Loadable Function: [u, s, v] = svd (a)

> Compute the singular value decomposition of a

>               A = U*S*V'

> The function `svd` normally returns the vector of singular values. If asked for 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

> If given a second argument, `svd` returns an economy-sized decomposition, eliminating the unnecessary rows or columns of u or v.

— Function File: [housv, beta, zer] = housh (x, j, z)

> Compute Householder reflection vector housv to reflect x to be the jth column of identity, i.e.,

>               (I - beta*housv*housv')x =  norm(x)*e(j) if x(1) < 0,
>               (I - beta*housv*housv')x = -norm(x)*e(j) if x(1) >= 0

> Inputs

> x
>   vector
> j
>   index into vector
> z
>   threshold for zero (usually should be the number 0)

> Outputs (see Golub and Van Loan):

> beta
>   If beta = 0, then no reflection need be applied (zer set to 0)
> housv
>   householder vector

— Function File: [u, h, nu] = krylov (a, v, k, eps1, pflg)

> Construct an orthogonal basis u of block Krylov subspace

>               [v a*v a^2*v ... a^(k+1)*v]

> Using 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: Hodel and Misra, "Partial Pivoting in the Computation of Krylov Subspaces", to be submitted to Linear Algebra and its Applications