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NAG[f08jkc] NAG[nag_dstein] - Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array
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Calling Sequence
f08jkc(d, e, w, iblock, isplit, z, ifailv, 'n'=n, 'm'=m, 'fail'=fail)
nag_dstein(. . .)
Parameters
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d - Vector(1..dim, datatype=float[8]);
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Note: the dimension, dim, of the array d must be at least  .
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On entry: the diagonal elements of the tridiagonal matrix  .
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e - Vector(1..dim, datatype=float[8]);
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Note: the dimension, dim, of the array e must be at least  .
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On entry: the off-diagonal elements of the tridiagonal matrix  .
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w - Vector(1..dim, datatype=float[8]);
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Note: the dimension, dim, of the array w must be at least  .
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iblock - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);
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Note: the dimension, dim, of the array iblock must be at least  .
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Constraint: ![iblock[i] <= iblock[i+1]](/support/helpjp/helpview.aspx?si=10441/file08260/math118.png) , for  . .
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isplit - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);
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Note: the dimension, dim, of the array isplit must be at least  .
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z - Matrix(1..dim1, 1..dim2, datatype=float[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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ifailv - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);
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Note: the dimension, dim, of the array ifailv must be at least  .
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On exit: if  , the first  elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to 0.
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'n'=n - integer; (optional)
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Default value: the dimension of the array d.
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On entry:  , the order of the matrix  .
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Constraint:  . .
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'm'=m - integer; (optional)
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Default value: the second dimension of the array z.
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On entry:  , the number of eigenvectors to be returned.
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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_dstein (f08jkc) computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration.
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Error Indicators and Warnings
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"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_BAD_PARAM"
On entry, argument  had an illegal value.
"NE_CONSTRAINT"
On entry,  , ![iblock[i]iblock[i+1]=< >](/support/helpjp/helpview.aspx?si=10441/file08260/math342.png) , ![w[i]w[i+1]=< >](/support/helpjp/helpview.aspx?si=10441/file08260/math344.png) . Constraint: if ![iblock[i] = iblock[i+1]](/support/helpjp/helpview.aspx?si=10441/file08260/math346.png) , ![w[i] <= w[i+1]](/support/helpjp/helpview.aspx?si=10441/file08260/math348.png) , for  .
"NE_CONVERGENCE"
 eigenvectors (as indicated by argument ifailv) each failed to converge in 5 iterations. The current iterate after 5 iterations is stored in the corresponding column of z.
"NE_INT"
On entry,  . Constraint:  .
On entry,  . Constraint:  .
"NE_INT_2"
On entry,  ,  . Constraint:  .
"NE_INTERNAL_ERROR"
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.
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Accuracy
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Each computed eigenvector ![z[i]](/support/helpjp/helpview.aspx?si=10441/file08260/math396.png) is the exact eigenvector of a nearby matrix ![A+E[i]](/support/helpjp/helpview.aspx?si=10441/file08260/math398.png) , such that
 = O(epsilon)*LinearAlgebra[Norm](A)](/support/helpjp/helpview.aspx?si=10441/file08260/math401.png)
where  is the machine precision. Hence the residual is small:
![||A,z[i],-,lambda[i],z[i]||=O(epsilon)||A||](/support/helpjp/helpview.aspx?si=10441/file08260/math407.png)
However, a set of eigenvectors computed by this function may not be orthogonal to so high a degree of accuracy as those computed by f08jec (nag_dsteqr).
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Examples
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n := 4:
m := 2:
d := Vector([2.07, 1.474093708197552, -0.6491595075457839, -1.694934200651768], datatype=float[8]):
e := Vector([-5.825753170191817, 2.624045178795587, 0.9162727563219193], datatype=float[8]):
w := Vector([-5.003353145492033, -1.998704574808668, 0, 0], datatype=float[8]):
iblock := Vector([1, 1, 0, 0], datatype=integer[kernelopts('wordsize')/8]):
isplit := Vector([4, 0, 0, 0], datatype=integer[kernelopts('wordsize')/8]):
z := Matrix(4, 2, datatype=float[8]):
ifailv := Vector(2, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f08jkc(d, e, w, iblock, isplit, z, ifailv, 'n' = n, 'm' = m):
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See Also
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Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Jessup E and Ipsen I C F (1992) Improving the accuracy of inverse iteration SIAM J. Sci. Statist. Comput. 13 550–572
f08 Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
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