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Cholesky factorization

returns an
upper triangular matrix `T`

= chol(`A`

)`T`

, such that `T'*T = A`

.
`A`

must be a Hermitian positive definite matrix. Otherwise, this syntax throws an error.

`[`

computes the Cholesky factorization of
`T`

,`p`

] =
chol(`A`

)`A`

. This syntax does not error if `A`

is not a
Hermitian positive definite matrix. If `A`

is a Hermitian positive
definite matrix, then `p`

is 0. Otherwise, `T`

is
`sym([])`

, and `p`

is a positive integer (typically,
`p = 1`

).

`___ = chol(`

skips checking whether matrix `A`

,`'nocheck'`

)`A`

is Hermitian positive definite.
`'nocheck'`

lets you compute Cholesky factorization of a matrix that
contains symbolic parameters without setting additional assumptions on those
parameters.

`___ = chol(`

computes the Cholesky factorization of `A`

,`'real'`

)`A`

using real arithmetic. In this
case, `chol`

computes a symmetric factorization ```
A =
T.'*T
```

instead of a Hermitian factorization `A = T'*T`

. This
approach is based on the fact that if `A`

is real and symmetric, then
`T'*T = T.'*T`

. Use `'real'`

to avoid complex conjugates
in the result.

`___ = chol(`

computes the Cholesky factorization of `A`

,`'lower'`

,`'nocheck'`

,`'real'`

)`A`

with one or more of these
optional arguments: `'lower'`

, `'nocheck'`

, and
`'real'`

. These optional arguments can appear in any order.

Compute the Cholesky factorization of the 3-by-3 Hilbert matrix. Because these numbers are not symbolic objects, you get floating-point results.

chol(hilb(3))

ans = 1.0000 0.5000 0.3333 0 0.2887 0.2887 0 0 0.0745

Now convert this matrix to a symbolic object, and compute the Cholesky factorization:

chol(sym(hilb(3)))

ans = [ 1, 1/2, 1/3] [ 0, 3^(1/2)/6, 3^(1/2)/6] [ 0, 0, 5^(1/2)/30]

Compute the Cholesky factorization of the 3-by-3 Pascal matrix returning a lower triangular matrix as a result:

chol(sym(pascal(3)), 'lower')

ans = [ 1, 0, 0] [ 1, 1, 0] [ 1, 2, 1]

Try to compute the Cholesky factorization of this matrix. Because
this matrix is not Hermitian positive definite, `chol`

used without
output arguments or with one output argument throws an error:

A = sym([1 1 1; 1 2 3; 1 3 5]);

T = chol(A)

Error using sym/chol (line 132) Cannot prove that input matrix is Hermitian positive definite. Define a Hermitian positive definite matrix by setting appropriate assumptions on matrix components, or use 'nocheck' to skip checking whether the matrix is Hermitian positive definite.

To suppress the error, use two output arguments, `T`

and
`p`

. If the matrix is not recognized as Hermitian positive definite, then
this syntax assigns an empty symbolic object to `T`

and the value
`1`

to `p`

:

[T,p] = chol(A)

T = [ empty sym ] p = 1

For a Hermitian positive definite matrix, `p`

is 0:

[T,p] = chol(sym(pascal(3)))

T = [ 1, 1, 1] [ 0, 1, 2] [ 0, 0, 1] p = 0

Alternatively, `'nocheck'`

lets you skip checking whether
`A`

is a Hermitian positive definite matrix. Thus, this flag lets you
compute the Cholesky factorization of a symbolic matrix without setting additional
assumptions on its components to make it Hermitian positive definite:

syms a A = [a 0; 0 a]; chol(A,'nocheck')

ans = [ a^(1/2), 0] [ 0, a^(1/2)]

If you use `'nocheck'`

for computing the Cholesky factorization of a
matrix that is not Hermitian positive definite, `chol`

can return a matrix
`T`

for which the identity `T'*T = A`

does not hold. To
make `isAlways`

return logical `0`

(`false`

) for undecidable conditions, set `Unknown`

to
`false`

.

T = chol(sym([1 1; 2 1]), 'nocheck')

T = [ 1, 2] [ 0, 3^(1/2)*1i]

isAlways(A == T'*T,'Unknown','false')

ans = 2×2 logical array 0 0 0 0

Compute the Cholesky factorization of the 3-by-3 inverse Hilbert matrix returning the permutation matrix:

A = sym(invhilb(3)); [T, p, S] = chol(A)

T = [ 3, -12, 10] [ 0, 4*3^(1/2), -5*3^(1/2)] [ 0, 0, 5^(1/2)] p = 0 S = 1 0 0 0 1 0 0 0 1

Compute the Cholesky factorization of the 3-by-3 inverse Hilbert matrix returning the permutation information as a vector:

A = sym(invhilb(3)); [T, p, S] = chol(A, 'vector')

T = [ 3, -12, 10] [ 0, 4*3^(1/2), -5*3^(1/2)] [ 0, 0, 5^(1/2)] p = 0 S = 1 2 3

Compute the Cholesky factorization of matrix `A`

containing symbolic parameters. Without additional assumptions on the parameter
`a`

, this matrix is not Hermitian. To make
`isAlways`

return logical `0`

(`false`

) for undecidable conditions, set `Unknown`

to
`false`

.

syms a A = [a 0; 0 a]; isAlways(A == A','Unknown','false')

ans = 2×2 logical array 0 1 1 0

By setting assumptions on `a`

and `b`

, you can define
`A`

to be Hermitian positive definite. Therefore, you can compute the
Cholesky factorization of `A`

:

assume(a > 0) chol(A)

ans = [ a^(1/2), 0] [ 0, a^(1/2)]

For further computations, remove the assumptions on `a`

by recreating
it using `syms`

:

syms a

Compute the Cholesky factorization of this matrix. To skip checking
whether it is Hermitian positive definite, use `'nocheck'`

. By default,
`chol`

computes a Hermitian factorization `A = T'*T`

.
Thus, the result contains complex conjugates.

syms a b A = [a b; b a]; T = chol(A, 'nocheck')

T = [ a^(1/2), conj(b)/conj(a^(1/2))] [ 0, (a*abs(a) - abs(b)^2)^(1/2)/abs(a)^(1/2)]

To avoid complex conjugates in the result, use `'real'`

:

T = chol(A, 'nocheck', 'real')

T = [ a^(1/2), b/a^(1/2)] [ 0, ((a^2 - b^2)/a)^(1/2)]

When you use this flag, `chol`

computes a symmetric factorization
`A = T.'*T`

instead of a Hermitian factorization ```
A =
T'*T
```

. To make `isAlways`

return logical `0`

(`false`

) for undecidable conditions, set `Unknown`

to
`false`

.

isAlways(A == T.'*T)

ans = 2×2 logical array 1 1 1 1

isAlways(A == T'*T,'Unknown','false')

ans = 2×2 logical array 0 0 0 0

Matrix computations involving many symbolic variables can be slow. To increase the computational speed, reduce the number of symbolic variables by substituting the given values for some variables.

Calling

`chol`

for numeric arguments that are not symbolic objects invokes the MATLAB^{®}`chol`

function.If you use

`'nocheck'`

, then the identities`T'*T = A`

(for an upper triangular matrix`T`

) and`T*T' = A`

(for a lower triangular matrix`T`

) are not guaranteed to hold.If you use

`'real'`

, then the identities`T'*T = A`

(for an upper triangular matrix`T`

) and`T*T' = A`

(for a lower triangular matrix`T`

) are only guaranteed to hold for a real symmetric positive definite`A`

.To use

`'vector'`

, you must specify three output arguments. Other flags do not require a particular number of output arguments.If you use

`'matrix'`

instead of`'vector'`

, then`chol`

returns permutation matrices, as it does by default.If you use

`'upper'`

instead of`'lower'`

, then`chol`

returns an upper triangular matrix, as it does by default.If

`A`

is not a Hermitian positive definite matrix, then the syntaxes containing the argument`p`

typically return`p = 1`

and an empty symbolic object`T`

.To check whether a matrix is Hermitian, use the operator

`'`

(or its functional form`ctranspose`

). Matrix`A`

is Hermitian if and only if`A'= A`

, where`A'`

is the conjugate transpose of`A`

.