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### 18.3 Matrix Factorizations

: R = chol (A)
: [R, p] = chol (A)
: [R, p, Q] = chol (A)
: [R, p, Q] = chol (A, "vector")
: [L, …] = chol (…, "lower")
: [R, …] = chol (…, "upper")

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.

: cholinv (A)

Compute the inverse of the symmetric positive definite matrix A using the Cholesky factorization.

: 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`.

: [R1, info] = cholupdate (R, u, op)

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

• R1’*R1 = R’*R + u*u’ if op is `"+"`
• R1’*R1 = R’*R - u*u’ if op is `"-"`

If op is `"-"`, info is set to

• 0 if the downdate was successful,
• 1 if R’*R - u*u’ is not positive definite,
• 2 if R is singular.

If info is not present, an error message is printed in cases 1 and 2.

: R1 = cholinsert (R, j, u)
: [R1, info] = cholinsert (R, j, u)

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

• 0 if the insertion was successful,
• 1 if A1 is not positive definite,
• 2 if R is singular.

If info is not present, an error message is printed in cases 1 and 2.

: R1 = choldelete (R, j)

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].

: R1 = cholshift (R, i, j)

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.

: H = hess (A)
: [P, H] = hess (A)

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).

See also: eig, chol, lu, qr, qz, schur, svd.

: [L, U] = lu (A)
: [L, U, P] = lu (A)
: [L, U, P, Q] = lu (S)
: [L, U, P, Q, R] = lu (S)
: […] = lu (S, thres)
: y = lu (…)
: […] = lu (…, "vector")

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`.

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 thres, that defines the pivoting threshold can be given. thres can be a scalar, in which case it defines the UMFPACK pivoting tolerance for both symmetric and unsymmetric cases. If thres 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.

See also: luupdate, ilu, chol, hess, qr, qz, schur, svd.

: [L, U] = luupdate (L, U, x, y)
: [L, U, P] = luupdate (L, U, P, x, y)

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.

: [Q, R] = qr (A)
: [Q, R, P] = qr (A) # non-sparse A
: X = qr (A)
: R = qr (A) # sparse A
: [C, R] = qr (A, B)
: […] = qr (…, 0)
: […] = qr (…, 'vector')
: […] = qr (…, 'matrix')

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
```

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).

If only 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 are requested.)

If the matrix A is full, 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)`.

If the matrix A is sparse, then the sparse QR factorization of A is computed using CSPARSE. As the matrix Q is in general a full matrix, it is recommended to request only one return value, which is the Q-less factorization R of A, such that `R = chol (A' * A)`.

If 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 the final argument is the scalar 0 and the number of rows is larger than the number of columns, then an `"economy"` factorization is returned, omitting zeroes of R and the corresponding columns of Q. That is, R will have only `size (A,1)` rows. In this case, P is a vector rather than a matrix.

If the final argument is the string `"vector"` then P is a permutation vector instead of a permutation matrix.

See also: chol, hess, lu, qz, schur, svd, qrupdate, qrinsert, qrdelete, qrshift.

: [Q1, R1] = qrupdate (Q, R, u, v)

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).

: [Q1, R1] = qrinsert (Q, R, j, x, orient)

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.

: [Q1, R1] = qrdelete (Q, R, j, orient)

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.

: [Q1, R1] = qrshift (Q, R, i, j)

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.

: lambda = qz (A, B)
: lambda = qz (A, B, opt)

QZ decomposition of the generalized eigenvalue problem (A x = s B x).

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 - s B)

```
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

`"+"`

= non-negative real part: leading block has all eigenvalues in the closed right half-plane

Note: `qz` performs permutation balancing, but not scaling (see XREFbalance). The order of output arguments was selected for compatibility with MATLAB.

See also: eig, balance, lu, chol, hess, qr, qzhess, schur, svd.

: [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.

See also: lu, chol, hess, qr, qz, schur, svd.

: S = schur (A)
: S = schur (A, "real")
: S = schur (A, "complex")
: S = schur (A, opt)
: [U, S] = schur (…)

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, lu, chol, hess, qr, qz, svd.

: [U, T] = rsf2csf (UR, TR)

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.

: [UR, SR] = ordschur (U, S, select)

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])
```

: angle = subspace (A, B)

Determine the largest principal angle between two subspaces spanned by the columns of matrices A and B.

: s = svd (A)
: [U, S, V] = svd (A)
: [U, S, V] = svd (A, econ)

Compute the singular value decomposition of A

```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
```

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

See also: svd_driver, svds, eig, lu, chol, hess, qr, qz.

: val = svd_driver ()
: old_val = svd_driver (new_val)
: svd_driver (new_val, "local")

Query or set the underlying LAPACK driver used by `svd`.

Currently recognized values are `"gesvd"` and `"gesdd"`. 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.

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

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

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

: [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: 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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