RNMVN
Generates pseudorandom numbers from a multivariate normal distribution.
Required Arguments
RSIG — Upper triangular matrix, K by K, containing the Cholesky factor of the variance‑covariance matrix. (Input)
The variance‑covariance matrix is equal to the product of the transpose of RSIG and RSIG. RSIG can be obtained from the variance‑covariance matrix using routine CHFAC.
R — NR by K matrix containing the random multivariate normal vectors in its rows. (Output)
Optional Arguments
NR — Number of random multivariate normal vectors to generate. (Input)
Default: NR = size (R,1).
K — Length of the multivariate normal vectors. (Input)
Default: K = size (R,2).
LDRSIG — Leading dimension of RSIG exactly as specified in the dimension statement in the calling program. (Input)
Default: LDRSIG = size (RSIG,1).
LDR — Leading dimension of R exactly as specified in the dimension statement of the calling program. (Input)
Default: LDR = size (R,1).
FORTRAN 90 Interface
Generic: CALL RNMVN (RSIG, R [, …])
Specific: The specific interface names are S_RNMVN and D_RNMVN.
FORTRAN 77 Interface
Single: CALL RNMVN (NR, K, RSIG, LDRSIG, R, LDR)
Double: The double precision name is DRNMVN.
Description
Routine RNMVN generates pseudorandom numbers from a multivariate normal distribution with mean vector consisting of all zeroes and variance‑covariance matrix whose Cholesky factor (or “square root”) is RSIG; that is, RSIG is an upper triangular matrix such that the transpose of RSIG times RSIG is the variance‑covariance matrix. First, independent random normal deviates with mean 0 and variance 1 are generated, and then the matrix containing these deviates is postmultiplied by RSIG. The independent normals are generated into the columns of a matrix, which has NR rows; hence, if RNSET is called with different values of NR, the output is different even if the seed is the same in the calls.
Deviates from a multivariate normal distribution with means other than zero can be generated by using RNMVN and then by adding the vector of means to each row of R.
Comments
The routine
RNSET can be used to initialize the seed of the random number generator. The routine
RNOPT can be used to select the form of the generator.
Example
In this example, RNMVN is used to generate five pseudorandom multivariate normal vectors of length 2 with variance‑covariance matrix equal to
The routine CHFAC is first called to compute the Cholesky factorization of the variance‑covariance matrix.
USE RNMVN_INT
USE UMACH_INT
USE CHFAC_INT
USE RNSET_INT
IMPLICIT NONE
INTEGER I, IRANK, ISEED, J, K, LDR, LDRSIG, NOUT, NR
REAL COV(2,2), R(5,2), RSIG(2,2)
!
CALL UMACH (2, NOUT)
NR = 5
K = 2
LDRSIG = 2
LDR = 5
COV(1,1) = 0.5
COV(1,2) = 0.375
COV(2,1) = 0.375
COV(2,2) = 0.5
! Obtain the Cholesky factorization.
CALL CHFAC (COV, IRANK, RSIG)
! Initialize seed of random number
! generator.
ISEED = 123457
CALL RNSET (ISEED)
CALL RNMVN (RSIG, R)
WRITE (NOUT,99999) ((R(I,J),J=1,K),I=1,NR)
99999 FORMAT (' Multivariate normal random deviates: ', /, &
(1X,2F8.4))
END
Output
Multivariate normal random deviates:
1.4507 1.2463
0.7660 -0.0429
0.0584 -0.6692
0.9035 0.4628
-0.8669 -0.9334
