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NAG[f02abc] NAG[nag_real_symm_eigensystem] - All eigenvalues and eigenvectors of real symmetric matrix
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Calling Sequence
f02abc(a, r, v, 'n'=n, 'tda'=tda, 'tdv'=tdv, 'fail'=fail)
nag_real_symm_eigensystem(. . .)
Parameters
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a - Matrix(1..n, 1..tda, datatype=float[8], order=C_order);
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r - Vector(1..n, datatype=float[8]);
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On exit: the eigenvalues in ascending order.
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v - Matrix(1..n, 1..tdv, datatype=float[8], order=C_order);
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On exit: the normalized eigenvectors, stored by columns; the  th column corresponds to the  th eigenvalue. The eigenvectors are normalized so that the sum of squares of the elements is equal to 1.
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'n'=n - integer; (optional)
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Default value: the first dimension of the arrays a, r, v and the second dimension of the arrays a, r, vthe arrays a, v.
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On entry:  , the order of the matrix  .
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Constraint:  . .
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'tda'=tda - integer; (optional)
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On entry: the second dimension of the array a as declared in the function from which nag_real_symm_eigensystem (f02abc) is called.
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Constraint:  . .
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'tdv'=tdv - integer; (optional)
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On entry: the second dimension of the array v as declared in the function from which nag_real_symm_eigensystem (f02abc) is called.
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Constraint:  . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_real_symm_eigensystem (f02abc) calculates all the eigenvalues and eigenvectors of a real symmetric matrix.
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Description
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nag_real_symm_eigensystem (f02abc) reduces the real symmetric matrix  to a real symmetric tridiagonal matrix by Householder's method. The eigenvalues and eigenvectors are calculated using the  algorithm.
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Error Indicators and Warnings
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"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_INT_ARG_LT"
On entry, n must not be less than 1:  .
"NE_TOO_MANY_ITERATIONS"
More than  iterations are required to isolate all the eigenvalues.
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Accuracy
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The eigenvectors are always accurately orthogonal but the accuracy of the individual eigenvectors is dependent on their inherent sensitivity to changes in the original matrix. For a detailed error analysis see pages 222 and 235 of Wilkinson and Reinsch (1971).
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Further Comments
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The time taken by nag_real_symm_eigensystem (f02abc) is approximately proportional to  .
The function may be called with the same actual array supplied for arguments a and v, in which case the eigenvectors will overwrite the original matrix.
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Examples
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>
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n := 4:
tda := 4:
tdv := 4:
a := Matrix([[0.5, 0, 2.3, -2.6], [0, 0.5, -1.4, -0.7], [2.3, -1.4, 0.5, 0], [-2.6, -0.7, 0, 0.5]], datatype=float[8], order='C_order'):
r := Vector(4, datatype=float[8]):
v := Matrix(4, 4, datatype=float[8], order='C_order'):
NAG:-f02abc(a, r, v, 'n' = n, 'tda' = tda, 'tdv' = tdv):
for i from 1 to 4 do
v[1..-1,i]:
end do:
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