f08fp computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

C#
public static void f08fp(
	string jobz,
	string range,
	string uplo,
	int n,
	Complex[,] a,
	double vl,
	double vu,
	int il,
	int iu,
	double abstol,
	out int m,
	double[] w,
	Complex[,] z,
	int[] jfail,
	out int info
)
Visual Basic
Public Shared Sub f08fp ( _
	jobz As String, _
	range As String, _
	uplo As String, _
	n As Integer, _
	a As Complex(,), _
	vl As Double, _
	vu As Double, _
	il As Integer, _
	iu As Integer, _
	abstol As Double, _
	<OutAttribute> ByRef m As Integer, _
	w As Double(), _
	z As Complex(,), _
	jfail As Integer(), _
	<OutAttribute> ByRef info As Integer _
)
Visual C++
public:
static void f08fp(
	String^ jobz, 
	String^ range, 
	String^ uplo, 
	int n, 
	array<Complex,2>^ a, 
	double vl, 
	double vu, 
	int il, 
	int iu, 
	double abstol, 
	[OutAttribute] int% m, 
	array<double>^ w, 
	array<Complex,2>^ z, 
	array<int>^ jfail, 
	[OutAttribute] int% info
)
F#
static member f08fp : 
        jobz : string * 
        range : string * 
        uplo : string * 
        n : int * 
        a : Complex[,] * 
        vl : float * 
        vu : float * 
        il : int * 
        iu : int * 
        abstol : float * 
        m : int byref * 
        w : float[] * 
        z : Complex[,] * 
        jfail : int[] * 
        info : int byref -> unit 

Parameters

jobz
Type: System..::..String
On entry: indicates whether eigenvectors are computed.
jobz="N"
Only eigenvalues are computed.
jobz="V"
Eigenvalues and eigenvectors are computed.
Constraint: jobz="N" or "V".
range
Type: System..::..String
On entry: if range="A", all eigenvalues will be found.
If range="V", all eigenvalues in the half-open interval vl,vu will be found.
If range="I", the ilth to iuth eigenvalues will be found.
Constraint: range="A", "V" or "I".
uplo
Type: System..::..String
On entry: if uplo="U", the upper triangular part of A is stored.
If uplo="L", the lower triangular part of A is stored.
Constraint: uplo="U" or "L".
n
Type: System..::..Int32
On entry: n, the order of the matrix A.
Constraint: n0.
a
Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,n
Note: the second dimension of the array a must be at least max1,n.
On entry: the n by n Hermitian matrix A.
  • If uplo="U", the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo="L", the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: the lower triangle (if uplo="L") or the upper triangle (if uplo="U") of a, including the diagonal, is overwritten.
vl
Type: System..::..Double
On entry: if range="V", the lower and upper bounds of the interval to be searched for eigenvalues.
If range="A" or "I", vl and vu are not referenced.
Constraint: if range="V", vl<vu.
vu
Type: System..::..Double
On entry: if range="V", the lower and upper bounds of the interval to be searched for eigenvalues.
If range="A" or "I", vl and vu are not referenced.
Constraint: if range="V", vl<vu.
il
Type: System..::..Int32
On entry: if range="I", the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range="A" or "V", il and iu are not referenced.
Constraints:
  • if range="I" and n=0, il=1 and iu=0;
  • if range="I" and n>0, 1iliun.
iu
Type: System..::..Int32
On entry: if range="I", the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range="A" or "V", il and iu are not referenced.
Constraints:
  • if range="I" and n=0, il=1 and iu=0;
  • if range="I" and n>0, 1iliun.
abstol
Type: System..::..Double
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b of width less than or equal to
abstol+εmaxa,b,
where ε is the machine precision. If abstol is less than or equal to zero, then εT1 will be used in its place, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2×x02am , not zero. If this method returns with info>0, indicating that some eigenvectors did not converge, try setting abstol to 2×x02am . See Demmel and Kahan (1990).
m
Type: System..::..Int32%
On exit: the total number of eigenvalues found. 0mn.
If range="A", m=n.
If range="I", m=iu-il+1.
w
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array w must be at least max1,n.
On exit: the first m elements contain the selected eigenvalues in ascending order.
z
Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint:
  • if jobz="V", dim1max1,n;
  • otherwise dim11.
Note: the second dimension of the array z must be at least max1,m if jobz="V", and at least 1 otherwise.
On exit: if jobz="V", then
  • if info=0, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with w[i-1];
  • if an eigenvector fails to converge (info>0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz="N", z is not referenced.
Note:  you must ensure that at least max1,m columns are supplied in the array z; if range="V", the exact value of m is not known in advance and an upper bound of at least n must be used.
jfail
Type: array<System..::..Int32>[]()[][]
An array of size [dim1]
Note: the dimension of the array jfail must be at least max1,n.
On exit: if jobz="V", then
  • if info=0, the first m elements of jfail are zero;
  • if info>0, jfail contains the indices of the eigenvectors that failed to converge.
If jobz="N", jfail is not referenced.
info
Type: System..::..Int32%
On exit: info=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDZ) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
If info=i, then i eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.
ifail=-9000
An error occured, see message report.
ifail=-6000
Invalid Parameters value
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value
ifail=-6000
Invalid Parameters value

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2=OεA2,
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

Parallelism and Performance

None.

Further Comments

The total number of floating-point operations is proportional to n3.
The real analogue of this method is f08fb.

Example

This example finds the eigenvalues in the half-open interval -2,2, and the corresponding eigenvectors, of the Hermitian matrix
A=12-i3-i4-i2+i23-2i4-2i3+i3+2i34-3i4+i4+2i4+3i4.

Example program (C#): f08fpe.cs

Example program data: f08fpe.d

Example program results: f08fpe.r

See Also