H — NH by NCOEF matrix H with each row corresponding to a row in the hypothesis and containing the constants that specify an estimable linear combination of the regression coefficients. (Input)

B — NCOEF by NDEP matrix

G — Matrix containing the null hypothesis values. (Input)
If NU = 0, then G is NH by NDEP; otherwise, G is NH by NU.

R — NCOEF by NCOEF upper triangular matrix containing the R matrix. (Input)
The R matrix can come from a regression fit based on a QR decomposition of the matrix of regressors or based on a Cholesky factorization RTR of the matrix of sums of squares and crossproducts of the regressors. Elements to the right of a diagonal element of R that is zero must also be zero. A zero row indicates a nonfull rank model. For an R matrix that comes from a regression fit with linear equality restrictions on the parameters, each row of R corresponding to a restriction must have a corresponding diagonal element that is negative. The remaining rows of R must have positive diagonal elements. Only the upper triangle of R is referenced.

SCPH — Matrix containing sums of squares and crossproducts attributable to the hypothesis. (Output)
If NU = 0, SCPH is a NDEP by NDEP matrix, otherwise, SCPH is a NU by NU matrix.

NH — Number of rows in the hypothesis. (Input)
Default: NH = size (H,1).

NCOEF — Number of regression coefficients in the model. (Input)
Default: NCOEF = size (H,2).

LDH — Leading dimension of H exactly as specified in the dimension statement of the calling program. (Input)
Default: LDH = size (H,1).

NDEP — Number of dependent (response) variables. (Input)
Default: NDEP = size (B,2).

LDB — Leading dimension of B exactly as specified in the dimension statement in the calling program. (Input)
Default: LDB = size (B,1).

NU — U matrix option. (Input)
For positive NU, NU is the number of linear combinations of the dependent variables to be considered. If NU = 0, the hypothesis is HB = G, i.e., U is automatically taken to be the identity. NU must be less than or equal to NDEP.
Default: NU = 0.

U — NDEP by NU matrix U in test H BU = G. (Input, if NU is positive)
If NU = 0, U is not referenced and can be a 1 x 1 array.
Default: U is a 1x 1 array.

LDU — Leading dimension of U exactly as specified in the dimension statement in the calling program. (Input)
Default: LDU = size (U, 1,).

LDG — Leading dimension of G exactly as specified in the dimension statement in the calling program. (Input)
Default: LDG = size (G,1).

LDR — Leading dimension of R exactly as specified in the dimension statement in the calling program. (Input)
Default: LDR = size (R,1).

DFH — Degrees of freedom for SCPH. (Output)
DFH equals the rank of H.

LDSCPH — Leading dimension of SCPH exactly as specified in the dimension statement in the calling program. (Input)
Default: LDSCPH = size (SCPH,1).

1. Form

2. Find C as the solution of RTC = HT using routine GIRTS (IMSL MATH/LIBRARY). If the equations are declared inconsistent within a computed tolerance, an error message with code 1 is issued that the hypothesis is not completely testable.

3. For all rows of R corresponding to restrictions, i.e., containing negative diagonal elements from a restricted least-squares fit using RLEQU, zero out the corresponding rows of C, i.e., form DC.

4. Decompose DC using Householder transformations and column pivoting to yield a square, upper triangular matrix T with diagonal elements of nonincreasing magnitude and permutation matrix P such that

5. Determine the rank of T, say r. If t11 = 0, then r = 0. Otherwise, the rank of T is r if

6. Find V as a solution to T TV = PTW using routine GIRTS. If the equations are inconsistent, an error message with code 2 is issued that the hypothesis is inconsistent within a computed tolerance, i.e., the linear system

7. Form VTV, which is the required matrix of sum of squares and crossproducts output in SCPH.

1. Workspace may be explicitly provided, if desired, by use of R2PSS/DR2PSS. The reference is:

2. Informational errors