mldivide
Lag operator polynomial left division
Syntax
B
= A
\C
B
= mldivide(A
, C
'PropertyName
',PropertyValue
)
Description
Given two lag operator polynomials, A(L) and C(L)
perform
a left division so that C(L) = A(L)*B(L),
or B(L)
= A(L)\C(L).
Left division requires invertibility of the coefficient matrix associated
with lag 0 of the denominator polynomial A(L).B
= A
\C
accepts
one or more commaseparated property name/value pairs.B
= mldivide(A
, C
'PropertyName
',PropertyValue
)
Input Arguments

Denominator (divisor) lag operator polynomial object, as produced
by 

Numerator (dividend) lag operator polynomial object, as produced
by If at least one of 

Nonnegative scalar absolute tolerance used as part of the termination
criterion of the calculation of the quotient coefficients and, subsequently,
to determine which coefficients to include in the quotient. Specifying
an absolute tolerance allows for customization of the termination
criterion. Once the algorithm has terminated, Default: 

Nonnegative scalar relative tolerance used as part of the termination
criterion of the calculation of the quotient coefficients. At each
lag, a coefficient matrix is calculated and its 2norm compared to
the largest coefficient 2norm. If the ratio of the current norm
to the largest norm is less than or equal to Default: 

Positive integer indicating the size of the window used to check
termination tolerances. Default: 

Nonnegative integer indicating the maximum degree of the quotient
polynomial. For stable denominators, the default is the power to which
the magnitude of the largest eigenvalue of the denominator must be
raised to equal the relative termination tolerance Default: 
Output Arguments

Quotient lag operator polynomial object, such that B(L) = A(L)\C(L). 
Examples
Tips
The right division operator (\) invokes mldivide
,
but the optional inputs are available only by calling mldivide
directly.
To rightinvert a stable B(L), set C(L) = eye
(B.Dimension
).
Algorithms
Lag operator polynomial division generally results in infinitedegree
polynomials. mldivide
imposes a termination criterion
to truncate the degree of the quotient polynomial.
If 'Degree'
is unspecified, the maximum degree
of the quotient is determined by the stability of the denominator.
Stable denominator polynomials usually result in quotients whose coefficients
exhibit geometric decay in absolute value. (When coefficients change
sign, it is the coefficient envelope which decays geometrically.)
Unstable denominators usually result in quotients whose coefficients
exhibit geometric growth in absolute value. In either case, maximum
degree will not exceed the value of 'Degree'
.
To control truncation error by terminating the coefficient sequence too early, the termination criterion involves three steps:
At each lag in the quotient polynomial, a coefficient matrix is calculated and tested against both a relative and an absolute tolerance (see
'RelTol'
and'AbsTol'
inputs ).If the current coefficient matrix is below either tolerance, then a tolerance window is opened to ensure that all subsequent coefficients remain below tolerance for a number of lags determined by
'Window'
.If any subsequent coefficient matrix within the window is above both tolerances, then the tolerance window is closed and additional coefficients are calculated, repeating steps (1) and (2) until a subsequent coefficient matrix is again below either tolerance, and a new window is opened.
Steps (1)(3) are repeated until a coefficient is below
tolerance and subsequent coefficients remains below tolerance for
'Window' lags, or until the maximum 'Degree'
is
encountered, or until a coefficient becomes numerically unstable (NaN
or +/Inf
).
References
[1] Box, G.E.P., G.M. Jenkins, and G.C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[2] Hayashi, F. Econometrics. Princeton, NJ: Princeton University Press, 2000.
[3] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.