inv, invpd#

Purpose#

inv() returns the inverse of an invertible matrix. invpd() returns the inverse of a symmetric, positive definite matrix.

Format#

xi = inv(x)#

xi = invpd(x)#

Parameters:: x (NxN matrix or K-dimensional array where the last two dimensions are NxN) – data
Returns:: xi (NxN matrix or K-dimensional array) – where the last two dimensions are NxN, containing the inverse of x.

Examples#

// Set random seed
rndseed 90701980;

// Set number of observations
n = 4000;

// Generate random x
x1 = rndn(n, 1);

// Add constant
x = ones(n, 1)~x1;

// Set true coefficients
btrue = { 1, 0.5 };

/*
** Generate linear data with
** random normal disturbances
*/
y = x*btrue + rndn(n, 1);

// Compute OLS estimates of coefficients
bols = invpd(x'x)*x'y;

After the code above, bols will be equal to:

1.0108262804
0.4633302971

This example simulates some data and computes the ols() coefficient estimator using the invpd() function. First, the number of observations is specified. Second, a vector x1 of standard Normal random variables is generated and is concatenated with a vector of ones() (to create a constant term). The true coefficients are specified, and the dependent variable y is created. Then the ols() coefficient estimates are computed.

When computing least-squares problems with poorly conditioned matrices, the slash operator / and the function olsqr() will provide greater accuracy.

Remarks#

x can be any legitimate expression that returns a matrix or array that is legal for the function.

If x is an array, the result will be an array containing the inverses of each 2-dimensional array described by the two trailing dimensions of x. In other words, for a 10x4x4 array, the result will be an array of the same size containing the inverses of each of the 10 4x4 arrays contained in x.

For inv(), if x is a matrix, it must be square and invertible. Otherwise, if x is an array, the 2-dimensional arrays described by the last two dimensions of x must be square and invertible.

For invpd(), if x is a matrix, it must be symmetric and positive definite. Otherwise, if x is an array, the 2-dimensional arrays described by the last two dimensions of x must be symmetric and positive definite.

If the input matrix is not invertible by these functions, they will either terminate the program with an error message or return an error code which can be tested for with the scalerr() function. This depends on the trap state as follows:

If trap is set to 1, they will return a scalar errorcode:

inv	invpd
50	20

If trap is set to 0, they will terminate with an error message:

inv	invpd
“Matrix singular”	“Matrix not positive definite”

If the input to invpd() is not symmetric, it is possible that the function will (erroneously) appear to operate successfully.

Positive definite matrices can be inverted by inv(). However, for symmetric, positive definite matrices (such as moment matrices), invpd() is about twice as fast as inv().