mldivide, \
Apply left division to lag operator polynomials
Description
The mldivide
function applies left division to input lag
operator polynomials, which are specified by LagOp
model objects (see Lag Operator Polynomial Left Division). To solve systems of linear
equations by applying left division to matrices, see mldivide
.
or
B
= A
\C
returns the quotient B
= mldivide(A
,C
)B
, the LagOp
lag operator polynomial resulting from the left division of the lag operator
polynomials
A
and C
.
specifies options using one or more name-value arguments. For example,
B
= mldivide(A
,C
,Name=Value
)mldivide(A,C,Degree=10,Window=5)
returns a quotient lag operator
polynomial with a maximum degree of 10
and terminates the division
algorithm when 5
consecutive lag coefficients satisfy the termination
tolerances.
Examples
Input Arguments
Name-Value Arguments
Output Arguments
More About
Tips
To left-invert a stable lag operator polynomial
A(L), specify the input A
as
A(L) and C
as
eye
(A.Dimension
).
Algorithms
Lag operator polynomial division can result in infinite-degree polynomials. mldivide
imposes a termination criterion to truncate the degree of
the quotient polynomial.
By default, the maximum degree of the quotient is determined by the stability of the
denominator (see Degree
). Stable denominator polynomials usually result
in quotients whose coefficients exhibit geometric decay in absolute value. (When coefficients
change sign, the coefficient envelope decays geometrically.) Unstable denominators usually
result in quotients whose coefficients exhibit geometric growth in absolute value. In either
case, Degree
specifies the maximum degree of the quotient.
To control truncation error by terminating the coefficient sequence too early, the termination criterion involves three steps:
At each lag in the quotient polynomial,
mldivide
calculates a coefficient matrix and tests it against both the specified relative and absolute tolerances, specified by theRelTol
andAbsTol
arguments.If the current coefficient matrix is below either tolerance,
mldivide
opens a tolerance window and ensures all subsequent coefficients remain below the tolerances for a number of lags specified by theWindow
argument.If any subsequent coefficient matrix within the window is above both tolerances,
mldivide
closes the tolerance window and calculates additional coefficients, iterating the process until a subsequent coefficient matrix is below either tolerance, causing a new tolerance window to open.
mldivide
repeats the process until a coefficient is
below tolerance and subsequent coefficients remain below tolerance for
Window
lags, until mldivide
produces
Degree
coefficients, or until a coefficient becomes numerically
unstable (has value NaN
or +/-Inf
).
References
[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[2] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Hayashi, Fumio. Econometrics. Princeton, NJ: Princeton University Press, 2000.
Version History
Introduced in R2010a