Forms Hadamard matrices (R.W. Payne).
Options
PRINT = string token |
Controls printed output (monitoring); default * i.e. none |
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METHOD = string token |
Method of construction (firstpaley, secondpaley, stored, sylvestre, tensorproduct, turyn, williamson); default * i.e. determined automatically |
Parameters
NROWS = scalars |
Number of rows of the matrices |
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HADAMARDMATRIX = matrices |
Saves the Hadamard matrices |
ERROR = scalars |
Returns 0 if the matrix has been formed successfully and 1 if not |
Description
A Hadamard matrix is a matrix containing values -1 and +1 whose rows are orthogonal: i.e. if H is a Hadamard matrix
H *+ T(H) = n * IDENTITY(n)
where n is the number of rows of H, which must be 2 or a multiple of 4.
FHADAMARDMATRIX provides several methods for forming the matrices, described in Hedayat, Sloane & Stufken (1999, Chapter 7). These work for all sizes of matrix up to 200, and various other sizes above that. The METHOD option can be used to specify the method. If this is unset FHADAMARDMATRIX selects an appropriate method automatically (you may then want to set the PRINT option to monitoring to record what method has been used). The settings of METHOD, in the order in which they will be selected, are as follows.
sylvestre |
uses a tensor product construction building on the 2 × 2 matrix to form matrices of size n = 2m for any positive integer m. |
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williamson |
builds the matrix from four circulant matrices; FHADAMARDMATRIX has a repertoire covering 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 148, 156 and 172 rows. |
turyn |
builds the matrix from a circulant and seven retrocirculant matrices; this method is used for the matrix with 188 rows. |
tensorproduct |
uses a tensor product between the matrix with two rows, and a Hadamard matrix with n/2 rows. |
secondpaley |
forms matrices with n = 2 × (s + 1), where s is a prime power, using properties of the Galois field of order s. |
firstpaley |
forms matrices with n = s + 1, where s is a prime power, using properties of the Galois field of order s. |
stored |
takes a matrix from a stored repertoire with rows from four to 32. |
The number of rows of the matrix is specified by the NROWS parameter, the HADAMARDMATRIX parameter saves the matrix, and the ERROR parameter can be set to a scalar which returns zero if the matrix has been formed successfully and one if not.
Options: PRINT, METHOD.
Parameters: NROWS, HADAMARDMATRIX, ERROR.
Method
The methods are described in Chapter 7 of Hedayat, Sloane & Stufken (1999).
Reference
Hedayat, A.S., Sloane, N.J.A., & Stufken, J. (1999). Orthogonal Arrays, Theory & Applications. Springer-Verlag, New York.
See also
Procedures: AGBIB, AGMAINEFFECT.
Commands for: Calculations and manipulation, Design of experiments.
Example
CAPTION 'FHADAMARDMATRIX example'; STYLE=meta SCALAR order; VALUE=8 FHADAMARD order; M PRINT M; FIELD=5; DECIMALS=1 " check matrix Hadamard " DIAGONAL [ROWS=order] I CALCULATE I = 1 MATRIX [ROWS=order; COLUMNS=order] MTM CALCULATE MTM = M*+T(M) & MTM= MTM - I*order & meancheck = MEAN(MTM) & varcheck = VAR(MTM) PRINT meancheck,varcheck