Chapter 2: Eigensystem Analysis

Routines

2.1.Eigenvalues and (Optionally) Eigenvectors of Ax = λx

2.1.4Real Band Symmetric Matrices in Band Storage Mode

2.1.6Real Upper Hessenberg Matrices

2.1.7Complex Upper Hessenberg Matrices

2.2.Eigenvalues and (Optionally) Eigenvectors of Ax = lBx

2.2.2Complex General Problem Ax = lBx

2.2.3Real Symmetric Problem Ax = lBx

Usage Notes

This chapter includes routines for linear eigensystem analysis. Many of these are for matrices with special properties. Some routines compute just a portion of the eigensystem. Use of the appropriate routine can substantially reduce computing time and storage requirements compared to computing a full eigensystem for a general complex matrix.

An ordinary linear eigensystem problem is represented by the equation Ax = lx where A denotes an n ´ n matrix. The value l is an eigenvalue and x ¹ 0 is the corresponding eigenvector. The eigenvector is determined up to a scalar factor. In all routines, we have chosen this factor so that x has Euclidean length with

value one, and the component of x of smallest index and largest magnitude is positive. In case x is a complex vector, this largest component is real and positive.

In contrast to Version 1.1 of the IMSL Libraries, in Version 2.0 the eigenvalues and corresponding eigenvectors are sorted and returned in the order of largest to smallest complex magnitude. Users who require access to specific eigenvalues may need to alter their application codes to adjust for this change. If the order returned by Version 1.1 eigensystem codes is required, a use of Level 2 routines can replace the use of Level 1 routines. The Level 2 routines are documented in Version 1.1.

For example, the single-precision routine for computing the complex eigenvalues of a non-symmetric real matrix should now be obtained with the statement:

CALL E2LRG (N,A,LDA,EVAL,ACOPY,RWK)

This will replace the Level 1 statement:

CALL EVLRG (N,A,LDA,EVAL)

The arrays A(*,*), EVAL(*), ACOPY(*), and RWK(*) are documented on page 293 of Version 1.1 (IMSL 1989).

Similar comments hold for the use of the remaining Level 1 routines in the following tables in those cases where the second character of the Level 2 routine name is no longer the character "2".

A generalized linear eigensystem problem is represented by Ax = λBx where A and B are n × n matrices. The value λ is an eigenvalue, and x is the corresponding eigenvector. The eigenvectors are normalized in the same manner as for the ordinary eigensystem problem. The linear eigensystem routines have names that begin with the letter “ E”. The generalized linear eigensystem routines have names that begin with the letter “ G”. This prefix is followed by a two-letter code for the type of analysis that is performed. That is followed by another two-letter suffix for the form of the coefficient matrix. The following tables summarize the names of the eigensystem routines.

Symmetric and Hermitian Eigensystems

Error Analysis and Accuracy

The remarks in this section are for the ordinary eigenvalue problem. Except in special cases, routines will not return the exact eigenvalue-eigenvector pair for the ordinary eigenvalue problem Ax = lx. The computed pair

~ ~

x , λ

is an exact eigenvector-eigenvalue pair for a “nearby” matrix A + E. Information about E is known only in terms of bounds of the form || E||2 £ ¦(n) ||A||2 e. The value of ¦(n) depends on the algorithm but is typically a small fractional power of n. The parameter e is the machine precision. By a theorem due to Bauer and Fike (see Golub and Van Loan [1989, page 342],

where s (A) is the set of all eigenvalues of A (called the spectrum of A), X is the matrix of eigenvectors, || × ||2 is the 2-norm, and k(X) is the condition number of X defined as k(X) = || X ||2 || X-1||2. If A is a real symmetric or complex Hermitian matrix, then its eigenvector matrix X is respectively orthogonal or unitary. For these matrices,k(X) = 1.

The eigenvalues

~

λ j

and eigenvectors

~

x j

computed by EVC** can be checked by computing their performance index t using EPI**. The performance index is defined by Smith et al. (1976, pages 124-126) to be

No significance should be attached to the factor of 10 used in the denominator. For a real vector x, the symbol || x ||1 represents the usual 1-norm of x. For a complex vector x, the symbol || x ||1 is defined by

N

x1 = å2 xk + xk 7

k =1

The performance index t is related to the error analysis because

where E is the “nearby” matrix discussed above.

While the exact value of τ is machine and precision dependent, the performance of an eigensystem analysis routine is defined as excellent if τ < 1, good if 1 ≤ τ ≤ 100, and poor if τ > 100. This is an arbitrary definition, but large values of τ can serve as a warning that there is a blunder in the calculation. There are also similar routines GPI** to compute the performance index for generalized eigenvalue problems.

If the condition number κ(X) of the eigenvector matrix X is large, there can be large errors in the eigenvalues even if τ is small. In particular, it is often difficult to recognize near multiple eigenvalues or unstable mathematical problems from numerical results. This facet of the eigenvalue problem is difficult to understand: A user often asks for the accuracy of an individual eigenvalue. This can be answered approximately by computing the condition number of an individual eigenvalue. See Golub and Van Loan (1989, pages 344-345). For matrices A such that the computed array of normalized eigenvectors X is invertible, the condition

number of λM is κM ≡ the Euclidean length of row j of the inverse matrix X-1 . Users can choose to compute this matrix with routine LINCG, page 43. An approximate bound for the accuracy of a computed eigenvalue is then given by κM ε || A || To compute an approximate bound for the relative accuracy of an eigenvalue, divide this bound by | λM |.

Reformulating Generalized Eigenvalue Problems

The generalized eigenvalue problem Ax = λBx is often difficult for users to analyze because it is frequently ill-conditioned. There are occasionally changes of variables that can be performed on the given problem to ease this ill-conditioning.

Suppose that B is singular but A is nonsingular. Define the reciprocal μ = λ-1. Then, the roles of A and B are interchanged so that the reformulated problem Bx = μAx is solved. Those generalized eigenvalues μM = 0 correspond to eigenvalues λM

= ∞. The remaining

λ j = μ−j 1

The generalized eigenvectors for λM correspond to those for μM. Other reformulations can be made: If B is nonsingular, the user can solve the ordinary eigenvalue problem Cx ≡ B-1 Ax = λx. This is not recommended as a computational algorithm for two reasons. First, it is generally less efficient than solving the generalized problem directly. Second, the matrix C will be subject to

perturbations due to ill-conditioning and rounding errors when computing B-1A. Computing the condition numbers of the eigenvalues for C may, however, be helpful for analyzing the accuracy of results for the generalized problem.

There is another method that users can consider to reduce the generalized problem to an alternate ordinary problem. This technique is based on first computing a matrix decomposition B = PQ, where both P and Q are matrices that are “simple” to invert. Then, the given generalized problem is equivalent

to the ordinary eigenvalue problem Fy = λy. The matrix F ≡ P-1AQ-1. The

unnormalized eigenvectors of the generalized problem are given by x = Q-1y. An example of this reformulation is used in the case where A and B are real and symmetric with B positive definite. The IMSL routines GVLSP, page 405 and

GVCSP, page 407, use P = R7 and Q = R where R is an upper triangular matrix

obtained from a Cholesky decomposition, B = R7R. The matrix F = R-T AR-1 is symmetric and real. Computation of the eigenvalue-eigenvector expansion for F is based on routine EVCSF, page 339.

EVLRG/DEVLRG (Single/Double precision)

Compute all of the eigenvalues of a real matrix.

Usage

CALL EVLRG (N, A, LDA, EVAL)

Arguments

N — Order of the matrix. (Input)

A — Real full matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

EVAL — Complex vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVLRG N(N + 6) units, or

DEVLRG 2N(N + 4) + 2n units.

Workspace may be explicitly provided, if desired, by use of

E3LRG/DE3LRG. The reference is

CALL E3LRG (N, A, LDA, EVAL, ACOPY, WK, IWK)

The additional arguments are as follows:

2.Informational error Type Code

41 The iteration for an eigenvalue failed to converge.

3.Integer Options with Chapter 10 Options Manager

1This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E3LRG, the internal or working leading dimension of ACOPY is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVLRG . Additional memory allocation and option value restoration are automatically done in EVLRG. There is no requirement that users change existing applications

that use EVLRG or E3LRG. Default values for the option are IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5 −8 in IVAL(*) are for

the generalized eigenvalue problem and are not used in EVLRG.

Algorithm

Routine EVLRG computes the eigenvalues of a real matrix. The matrix is first balanced. Elementary or Gauss similarity transformations with partial pivoting are

used to reduce this balanced matrix to a real upper Hessenberg matrix. A hybrid double−shifted LR−QR algorithm is used to compute the eigenvalues of the

Hessenberg matrix, Watkins and Elsner (1990).

The balancing routine is based on the EISPACK routine BALANC. The reduction routine is based on the EISPACK routine ELMHES. See Smith et al. (1976) for the EISPACK routines. The LR−QR algorithm is based on software work of Watkins and Haag. Further details, some timing data, and credits are given in Hanson et al. (1990).

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 85). The eigenvalues of this real matrix are computed and printed. The exact eigenvalues are known to be {4, 3, 2, 1}.

Output

EVCRG/DEVCRG (Single/Double precision)

Compute all of the eigenvalues and eigenvectors of a real matrix.

Usage

CALL EVCRG (N, A, LDA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix. (Input)

A — Floating-point array containing the matrix. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

EVAL — Complex array of size N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Complex array containing the matrix of eigenvectors. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

1.Automatic workspace usage is

EVCRG 2N * (N + 1) + 8N units, or

DEVCRG 4N * (N + 1) + 13N + N

Workspace may be explicitly provided, if desired, by use of E8CRG/DE8CRG. The reference is:

CALL E8CRG (N, A, LDA, EVAL, EVEC, LDEVEC, ACOPY, ECOPY WK,IWK)

The additional arguments are as follows:

ACOPY — Floating-point work array of size N by N. The arrays A and

ACOPY may be the same, in which case the first N2 elements of A will be destroyed. The array ACOPY can have its working row dimension increased from N to a larger value. An optional usage is required. See Item 3 below for further details.

ECOPY — Floating-point work array of default size N by N + 1. The working, leading dimension of ECOPY is the same as that for ACOPY. To increase this value, an optional usage is required. See Item 3 below for further details.

WK — Floating-point work array of size 6 N.

IWK — Integer work array of size N.

2.Informational error Type Code

41 The iteration for the eigenvalues failed to converge. No eigenvalues or eigenvectors are computed.

3.Integer Options with Chapter 10 Options Manager

1This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E8CRG, the internal or working leading dimensions of ACOPY and ECOPY are both increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVCRG. Additional memory allocation and option value restoration are automatically done in EVCRG. There is no requirement that users change existing applications that use EVCRG or E8CRG.

Default values for the option are IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5−8 in IVAL(*) are for the generalized eigenvalue

problem and are not used in EVCRG.

Algorithm

Routine EVCRG computes the eigenvalues and eigenvectors of a real matrix. The

matrix is first balanced. Orthogonal similarity transformations are used to reduce the balanced matrix to a real upper Hessenberg matrix. The implicit double−

shifted QR algorithm is used to compute the eigenvalues and eigenvectors of this Hessenberg matrix. The eigenvectors are normalized such that each has Euclidean length of value one. The largest component is real and positive.

The balancing routine is based on the EISPACK routine BALANC. The reduction routine is based on the EISPACK routines ORTHES and ORTRAN. The QR algorithm routine is based on the EISPACK routine HQR2. See Smith et al.

(1976) for the EISPACK routines. Further details, some timing data, and credits are given in Hanson et al. (1990).

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 82). The eigenvalues and eigenvectors of this real matrix are computed and printed. The performance index is also computed and printed. This serves as a check on the computations. For more details, see IMSL routine EPIRG, page 330.

CALL WRCRN (’EVEC’, N, N, EVEC, LDEVEC, 0)

WRITE (NOUT,’(/,A,F6.3)’) ’ Performance index = ’, PI

END

Output

EPIRG/DEPIRG (Single/Double precision)

Compute the performance index for a real eigensystem.

Usage

EPIRG(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

NEVAL — Number of eigenvalue/eigenvector pairs on which the performance index computation is based. (Input)

A — Matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Complex vector of length NEVAL containing eigenvalues of A. (Input)

EVEC — Complex N by NEVAL array containing eigenvectors of A. (Input) The eigenvector corresponding to the eigenvalue EVAL(J) must be in the J-th column of EVEC.

EPIRG — Performance index. (Output)

Comments

1.Automatic workspace usage is

EPIRG 2N units, or

DEPIRG 4N units.

Workspace may be explicitly provided, if desired, by use of E2IRG/DE2IRG. The reference is

E2IRG(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC, CWK)

The additional argument is

CWK — Complex work array of length N.

2.Informational errors

Algorithm

Let M = NEVAL, λ = EVAL, xM = EVEC(*,J), the j-th column of EVEC. Also, let ε be the machine precision given by AMACH(4). The performance index, τ, is defined to be

The norms used are a modified form of the 1-norm. The norm of the complex vector v is

N

v1 = å< vi + vi A

i=1

While the exact value of τ is highly machine dependent, the performance of EVCSF is considered excellent if τ < 1, good if 1 ≤ τ ≤ 100, and poor if τ > 100.

The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124−125).

Example

For an example of EPIRG, see IMSL routine EVCRG, page 327.

EVLCG/DEVLCG (Single/Double precision)

Compute all of the eigenvalues of a complex matrix.

Usage

CALL EVLCG (N, A, LDA, EVAL)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Complex vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVLCG 2N2 + 6N units, or

DEVLCG 4N2 + 11N units.

Workspace may be explicitly provided, if desired, by use of

E3LCG/DE3LCG. The reference is

CALL E3LCG (N, A, LDA, EVAL, ACOPY, RWK, CWK, IWK)

The additional arguments are as follows:

2.Informational error Type Code

41 The iteration for an eigenvalue failed to converge.

3.Integer Options with Chapter 10 Options Manager

1This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E3LCG, the internal or working, leading dimension of ACOPY is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL (4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVLCG . Additional memory allocation and option value restoration are automatically done in EVLCG. There is no requirement that users change existing

applications that use EVLCG or E3LCG. Default values for the option are IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5 −8 in

IVAL(*) are for the generalized eigenvalue problem and are not used in EVLCG.

Algorithm

Routine EVLCG computes the eigenvalues of a complex matrix. The matrix is first balanced. Unitary similarity transformations are used to reduce this balanced matrix to a complex upper Hessenberg matrix. The shifted QR algorithm is used to compute the eigenvalues of this Hessenberg matrix.

The balancing routine is based on the EISPACK routine CBAL. The reduction routine is based on the EISPACK routine CORTH. The QR routine used is based on the EISPACK routine COMQR2. See Smith et al. (1976) for the EISPACK routines.

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 115). The program computes the eigenvalues of this matrix.

DATA A/(1.0,2.0), (43.0,44.0), (5.0,6.0), (3.0,4.0),

&(13.0,14.0), (7.0,8.0), (21.0,22.0), (15.0,16.0),

&(25.0,26.0)/

Output

EVCCG/DEVCCG (Single/Double precision)

Compute all of the eigenvalues and eigenvectors of a complex matrix.

Usage

CALL EVCCG (N, A, LDA, EVAL, EVEC, LDEVEC)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Complex vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Complex matrix of order N. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

1.Automatic workspace usage is

EVCCG 2N2 + 6N units, or

DEVCCG 4N2 + 11N units.

Workspace may be explicitly provided, if desired, by use of E6CCG/DE6CCG. The reference is

CALL E6CCG (N, A, LDA, EVAL, EVEC, LDEVEC, ACOPY, RWK, CWK, IWK)

The additional arguments are as follows:

2.Informational error Type Code

41 The iteration for the eigenvalues failed to converge. No eigenvalues or eigenvectors are computed.

3.Integer Options with Chapter 10 Options Manager

1This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E6CCG, the internal or working leading dimensions of ACOPY and ECOPY are both increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVCCG. Additional memory allocation and option value restoration are automatically done in EVCCG. There is no requirement that users change existing applications that use EVCCG or E6CCG.

Default values for the option are IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5−8 in IVAL(*) are for the generalized eigenvalue

problem and are not used in EVCCG.

Algorithm

Routine EVCCG computes the eigenvalues and eigenvectors of a complex matrix. The matrix is first balanced. Unitary similarity transformations are used to reduce this balanced matrix to a complex upper Hessenberg matrix. The QR algorithm is used to compute the eigenvalues and eigenvectors of this Hessenberg matrix. The eigenvectors of the original matrix are computed by transforming the eigenvectors of the complex upper Hessenberg matrix.

The balancing routine is based on the EISPACK routine CBAL. The reduction routine is based on the EISPACK routine CORTH. The QR algorithm routine used is based on the EISPACK routine COMQR2. The back transformation routine is based on the EISPACK routine CBABK2 . See Smith et al. (1976) for the EISPACK routines.

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 116). Its eigenvalues are known to be {1 + 5i, 2 + 6i, 3 + 7i, 4 + 8i}. The program computes the eigenvalues and eigenvectors of this matrix. The performance index is also computed and printed. This serves as a check on the computations; for more details, see IMSL routine EPICG, page 336.

DATA A/(5.0,9.0), (3.0,3.0), (2.0,2.0), (1.0,1.0), (5.0,5.0),

&(6.0,10.0), (3.0,3.0), (2.0,2.0), (-6.0,-6.0), (-5.0,-5.0),

&(-1.0,3.0), (-3.0,-3.0), (-7.0,-7.0), (-6.0,-6.0),

&(-5.0,-5.0), (0.0,4.0)/

WRITE (NOUT,’(/,A,F6.3)’) ’ Performance index = ’, PI

END

Output

EPICG/DEPICG (Single/Double precision)

Compute the performance index for a complex eigensystem.

Usage

EPICG(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

NEVAL — Number of eigenvalue/eigenvector pairs on which the performance index computation is based. (Input)

A — Complex matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Complex vector of length N containing the eigenvalues of A. (Input)

EVEC — Complex matrix of order N containing the eigenvectors of A. (Input) The J-th eigenvalue/eigenvector pair should be in EVAL(J) and in the J-th column of EVEC.

LDEVEC — Leading dimension of EVEC exactly as specified in the dimension statement in the calling program. (Input)

EPICG — Performance index. (Output)

Comments

1.Automatic workspace usage is

EPICG 2N units, or

DEPICG 4N units.

Workspace may be explicitly provided, if desired, by use of E2ICG/DE2ICG. The reference is

E2ICG(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC, WK)

The additional argument is

WK — Complex work array of length N.

2.Informational errors

Algorithm

Let M = NEVAL, λ = EVAL, xM = EVEC(*, J), the j-th column of EVEC. Also, let ε be the machine precision given by AMACH(4). The performance index, τ, is defined to be

The norms used are a modified form of the 1-norm. The norm of the complex vector v is

N

v1 = å< vi + vi A

i=1

While the exact value of τ is highly machine dependent, the performance of EVCSF (page 339) is considered excellent if τ < 1, good if 1 ≤ τ ≤ 100, and poor

if τ > 100. The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124−125).

Example

For an example of EPICG, see IMSL routine EVCCG on page 333.

EVLSF/DEVLSF (Single/Double precision)

Compute all of the eigenvalues of a real symmetric matrix.

Usage

CALL EVLSF (N, A, LDA, EVAL)

Arguments

N — Order of the matrix A. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Real vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVLSF 3N units, or

DEVLSF 4N + N units.

Workspace may be explicitly provided, if desired, by use of E4LSF/DE4LSF. The reference is

CALL E4LSF (N, A, LDA, EVAL,WORK, IWORK)

31 The iteration for the eigenvalue failed to converge in 100 iterations before deflating.

Algorithm

Routine EVLSF computes the eigenvalues of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. Then, an implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix.

The reduction routine is based on the EISPACK routine TRED2. See Smith et al. (1976). The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169). Further details, some timing data, and credits are given in Hanson et al. (1990).

Example

In this example, the eigenvalues of a real symmetric matrix are computed and printed. This matrix is given by Gregory and Karney (1969, page 56).

Output

15.005.00 5.00 -1.00

EVCSF/DEVCSF (Single/Double precision)

Compute all of the eigenvalues and eigenvectors of a real symmetric matrix.

Usage

CALL EVCSF (N, A, LDA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Real vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Real matrix of order N. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

1.Automatic workspace usage is

EVCSF 4N units, or

DEVCSF 7N units.

Workspace may be explicitly provided, if desired, by use of E5CSF/DE5CSF. The reference is

CALL E5CSF (N, A, LDA, EVAL, EVEC, LDEVEC, WORK,

IWK)

The additional argument is

WORK — Work array of length 3 N.

IWK — Integer array of length N.

2.Informational error

Type Code

31 The iteration for the eigenvalue failed to converge in 100 iterations before deflating.

Algorithm

Routine EVCSF computes the eigenvalues and eigenvectors of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. These transformations are accumulated. An implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix. The eigenvectors are computed using the eigenvalues as perfect shifts, Parlett (1980, pages 169, 172). The reduction routine is based on the EISPACK routine TRED2. See Smith et al. (1976) for the EISPACK routines. Further details, some timing data, and credits are given in Hanson et al. (1990).

Example

The eigenvalues and eigenvectors of this real symmetric matrix are computed and printed. The performance index is also computed and printed. This serves as a check on the computations. For more details, see EPISF on page 350.

CALL WRRRN (’EVEC’, N, N, EVEC, LDEVEC, 0)

WRITE (NOUT, ’(/,A,F6.3)’) ’ Performance index = ’, PI

END

Output

Performance index = 0.044

EVASF/DEVASF (Single/Double precision)

Compute the largest or smallest eigenvalues of a real symmetric matrix.

Usage

CALL EVASF (N, NEVAL, A, LDA, SMALL, EVAL)

Arguments

N — Order of the matrix A. (Input)

NEVAL — Number of eigenvalues to be computed. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

SMALL — Logical variable. (Input)

If .TRUE., the smallest NEVAL eigenvalues are computed. If .FALSE., the largest NEVAL eigenvalues are computed.

EVAL — Real vector of length NEVAL containing the eigenvalues of A in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVASF 5N units, or

DEVASF 9N units.

Workspace may be explicitly provided, if desired, by use of E4ASF/DE4ASF. The reference is

CALL E4ASF (N, NEVAL, A, LDA, SMALL, EVAL, WORK,

IWK)

WORK — Work array of length 4 N.

IWK — Integer work array of length N.

2.Informational error

EVESF/DEVESF (Single/Double precision)

Compute the largest or smallest eigenvalues and the corresponding eigenvectors of a real symmetric matrix.

Usage

CALL EVESF (N, NEVEC, A, LDA, SMALL, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

NEVEC — Number of eigenvalues to be computed. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

SMALL — Logical variable. (Input)

If .TRUE., the smallest NEVEC eigenvalues are computed. If .FALSE., the largest NEVEC eigenvalues are computed.

EVAL — Real vector of length NEVEC containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Real matrix of dimension N by NEVEC. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

1.Automatic workspace usage is

EVESF 10N units, or DEVESF 19N units.

Workspace may be explicitly provided, if desired, by use of E5ESF/DE5ESF. The reference is

CALL E5ESF (N, NEVEC, A, LDA, SMALL, EVAL, EVEC, LDEVEC, WK, IWK)

The additional arguments are as follows:

2.Informational errors

Type Code

31 The iteration for an eigenvalue failed to converge. The best estimate will be returned.

32 Inverse iteration did not converge. Eigenvector is not correct for the specified eigenvalue.

Algorithm

Routine EVESF computes the largest or smallest eigenvalues and the corresponding eigenvectors of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. Then, an implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix. Inverse iteration is used to compute the eigenvectors of the tridiagonal matrix. This is followed by orthogonalization of these vectors. The eigenvectors of the original matrix are computed by back transforming those of the tridiagonal matrix.

The reduction routine is based on the EISPACK routine TRED2. See Smith et al. (1976). The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169). The inverse iteration and orthogonalization computation is discussed in Hanson et al. (1990). The back transformation routine is based on the EISPACK routine TRBAK1.

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 55). The largest two eigenvalues and their eigenvectors are computed and printed. The performance index is also computed and printed. This serves as a check on the computations. For more details, see IMSL routine EPISF on page 350.

NEVEC = 2

SMALL = .FALSE.

WRITE (NOUT,’(/,A,F6.3)’) ’ Performance index = ’, PI

END

Output

EVAL

1 2

10.005.00

EVEC

1 2

10.6325 -0.3162

20.6325 -0.3162

Performance index = 0.026

EVBSF/DEVBSF (Single/Double precision)

Compute selected eigenvalues of a real symmetric matrix.

Usage

CALL EVBSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, EVAL)

Arguments

N — Order of the matrix A. (Input)

MXEVAL — Maximum number of eigenvalues to be computed. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

ELOW — Lower limit of the interval in which the eigenvalues are sought. (Input)

EHIGH — Upper limit of the interval in which the eigenvalues are sought. (Input)

NEVAL — Number of eigenvalues found. (Output)

EVAL — Real vector of length MXEVAL containing the eigenvalues of A in the interval (ELOW, EHIGH) in decreasing order of magnitude. (Output)

Only the first NEVAL elements of EVAL are significant.

Comments

1.Automatic workspace usage is

EVBSF 6N units, or DEVBSF 11N units.

Workspace may be explicitly provided, if desired, by use of E5BSF/DE5BSF. The reference is

CALL E5BSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL,

EVAL, WK, IWK)

The additional arguments are as follows:

WK — Work array of length 5 N.

IWK — Integer work array of length 1 N.

2.Informational error

Type Code

31 The number of eigenvalues in the specified interval exceeds MXEVAL. NEVAL contains the number of eigenvalues in the interval. No eigenvalues will be returned.

Algorithm

Routine EVBSF computes the eigenvalues in a given interval for a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. Then, an implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix. The reduction step is based on the EISPACK routine TRED1. See Smith et al. (1976). The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169).

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 56). The eigenvalues of A are known to be −1, 5, 5 and 15. The eigenvalues in the interval [1.5, 5.5] are computed and printed. As a test, this example uses MXEVAL = 4. The routine EVBSF computes NEVAL, the number of eigenvalues in the given interval. The value of NEVAL is 2.

Output

EVFSF/DEVFSF (Single/Double precision)

Compute selected eigenvalues and eigenvectors of a real symmetric matrix.

Usage

CALL EVFSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, EVAL,

EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

MXEVAL — Maximum number of eigenvalues to be computed. (Input)

A — Real symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

ELOW — Lower limit of the interval in which the eigenvalues are sought. (Input)

EHIGH — Upper limit of the interval in which the eigenvalues are sought. (Input)

NEVAL — Number of eigenvalues found. (Output)

EVAL — Real vector of length MXEVAL containing the eigenvalues of A in the interval (ELOW, EHIGH) in decreasing order of magnitude. (Output)

Only the first NEVAL elements of EVAL are significant.

EVEC — Real matrix of dimension N by MXEVAL. (Output)

The J-th eigenvector corresponding to EVAL(J), is stored in the J-th column. Only the first NEVAL columns of EVEC are significant. Each vector is normalized to have Euclidean length equal to the value one.

1.Automatic workspace usage is

EVFSF 10N units, or DEVFSF 18N + N units.

Workspace may be explicitly provided, if desired, by use of E3FSF/DE3FSF. The reference is

CALL E3FSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, EVAL, EVEC, LDEVEC, WK, IWK)

The additional arguments are as follows:

WK — Work array of length 9 N.

IWK — Integer work array of length N.

2.Informational errors

Type Code

31 The number of eigenvalues in the specified range exceeds MXEVAL. NEVAL contains the number of eigenvalues in the range. No eigenvalues will be

computed.

32 Inverse iteration did not converge. Eigenvector is not correct for the specified eigenvalue.

Algorithm

Routine EVFSF computes the eigenvalues in a given interval and the corresponding eigenvectors of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. Then, an implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix. Inverse iteration is used to compute the eigenvectors of the tridiagonal matrix. This is followed by orthogonalization of these vectors. The eigenvectors of the original matrix are computed by back transforming those of the tridiagonal matrix.

The reduction step is based on the EISPACK routine TRED1. The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169). The inverse iteration and orthogonalization processes are discussed in Hanson et al. (1990). The transformation back to the users’s input matrix is based on the EISPACK routine TRBAK1. See Smith et al. (1976) for the EISPACK routines.

Example

In this example, A is set to the computed Hilbert matrix. The eigenvalues in the interval [0.001, 1] and their corresponding eigenvectors are computed and printed. This example uses MXEVAL = 3. The routine EVFSF computes the number of eigenvalues NEVAL in the given interval. The value of NEVAL is 2. The performance index is also computed and printed. For more details, see IMSL routine EPISF on page 350.

&EVEC(LDEVEC,MXEVAL), PI

CALL EVFSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, EVAL, EVEC,

&LDEVEC)

Output

3 0.6490 -0.6887

Performance index = 0.008

EPISF/DEPISF (Single/Double precision)

Compute the performance index for a real symmetric eigensystem.

Usage

EPISF(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

NEVAL — Number of eigenvalue/eigenvector pairs on which the performance index computation is based on. (Input)

A — Symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

EVAL — Vector of length NEVAL containing eigenvalues of A. (Input)

EVEC — N by NEVAL array containing eigenvectors of A. (Input)

The eigenvector corresponding to the eigenvalue EVAL(J) must be in the J-th column of EVEC.

EPISF — Performance index. (Output)

Comments

1.Automatic workspace usage is

EPISF N units, or

DEPISF 2N units.

Workspace may be explicitly provided, if desired, by use of E2ISF/DE2ISF. The reference is

E2ISF(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC, WORK)

The additional argument is

WORK — Work array of length N.

E2ISF — Performance Index.

2.Informational errors

Algorithm

Let M = NEVAL, λ = EVAL, xM = EVEC(*,J), the j-th column of EVEC. Also, let ε be the machine precision, given by AMACH(4) (page 1201). The performance index, τ, is defined to be

While the exact value of τ is highly machine dependent, the performance of EVCSF (page 339) is considered excellent if τ < 1, good if 1 ≤ τ ≤ 100, and poor

if τ > 100. The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124−125).

Example

For an example of EPISF, see routine EVCSF, page 339.

EVLSB/DEVLSB (Single/Double precision)

Compute all of the eigenvalues of a real symmetric matrix in band symmetric storage mode.

Usage

CALL EVLSB (N, A, LDA, NCODA, EVAL)

Arguments

N — Order of the matrix A. (Input)

A — Band symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

NCODA — Number of codiagonals in A. (Input)

EVAL — Vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVLSB N(NCODA + 2) units, or

DEVLSB 2N(NCODA + 2) units.

Workspace may be explicitly provided, if desired, by use of

E3LSB/DE3LSB. The reference is

CALL E3LSB (N, A, LDA, NCODA, EVAL, ACOPY, WK)

The additional arguments are as follows:

ACOPY — Work array of length N(NCODA + 1). The arrays A and ACOPY may be the same, in which case the first N(NCODA + 1) elements of A will be destroyed.

WK — Work array of length N.

2.Informational error

Algorithm

Routine EVLSB computes the eigenvalues of a real band symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. The implicit QL algorithm is used to compute the eigenvalues of the resulting tridiagonal matrix.

The reduction routine is based on the EISPACK routine BANDR; see Garbow et al. (1977). The QL routine is based on the EISPACK routine IMTQL1; see Smith et al. (1976).

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney (1969, page 77). The eigenvalues of this matrix are given by

Since the eigenvalues returned by EVLSB are in decreasing magnitude, the above formula for k = 1, ¼, N gives the the values in a different order. The eigenvalues of this real band symmetric matrix are computed and printed.

Output

EVCSB/DEVCSB (Single/Double precision)

Compute all of the eigenvalues and eigenvectors of a real symmetric matrix in band symmetric storage mode.

Usage

CALL EVCSB (N, A, LDA, NCODA, EVAL, EVEC, LDEVEC)

Arguments

N — Order of the matrix A. (Input)

A — Band symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

NCODA — Number of codiagonals in A. (Input)

EVAL — Vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Matrix of order N containing the eigenvectors. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

Comments

1.Automatic workspace usage is

EVCSB N(NCODA + 3) units, or

DEVCSB 2N(NCODA + 2) + N units.

Workspace may be explicitly provided, if desired, by use of

E4CSB/DE4CSB. The reference is

CALL E4CSB (N, A, LDA, NCODA, EVAL, EVEC, LDEVEC, ACOPY, WK,IWK)

The additional arguments are as follows:

ACOPY — Work array of length N(NCODA + 1). A and ACOPY may be the same, in which case the first N * NCODA elements of A will be destroyed.

WK — Work array of length N.

IWK — Integer work array of length N.

2.Informational error Type Code

41 The iteration for the eigenvalues failed to converge.

3.The success of this routine can be checked using EPISB (page 366).

Algorithm

Routine EVCSB computes the eigenvalues and eigenvectors of a real band symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. These transformations are accumulated. The implicit QL algorithm is used to compute the eigenvalues and eigenvectors of the resulting tridiagonal matrix.

The reduction routine is based on the EISPACK routine BANDR; see Garbow et al. (1977). The QL routine is based on the EISPACK routine IMTQL2; see Smith et al. (1976).

Example

In this example, a DATA statement is used to set A to a band matrix given by Gregory and Karney (1969, page 75). The eigenvalues, λN, of this matrix are given by

The eigenvalues and eigenvectors of this real band symmetric matrix are computed and printed. The performance index is also computed and printed. This serves as a check on the computations; for more details, see IMSL routine EPISB, page 366.

Output

EVASB/DEVASB (Single/Double precision)

Compute the largest or smallest eigenvalues of a real symmetric matrix in band symmetric storage mode.

Usage

CALL EVASB (N, NEVAL, A, LDA, NCODA, SMALL, EVAL)

Arguments

N — Order of the matrix A. (Input)

NEVAL — Number of eigenvalues to be computed. (Input)

A — Band symmetric matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

NCODA — Number of codiagonals in A. (Input)

SMALL — Logical variable. (Input)

If .TRUE., the smallest NEVAL eigenvalues are computed. If .FALSE., the largest NEVAL eigenvalues are computed.

EVAL — Vector of length NEVAL containing the computed eigenvalues in decreasing order of magnitude. (Output)

Comments

1.Automatic workspace usage is

EVASB N(NCODA + 4) units, or

DEVASB 2N(NCODA + 4) units.

Workspace may be explicitly provided, if desired, by use of

E3ASB/DE3ASB. The reference is

CALL E3ASB (N, NEVAL, A, LDA, NCODA, SMALL, EVAL,

ACOPY, WK)

The additional arguments are as follows:

ACOPY — Work array of length N(NCODA + 1). A and ACOPY may be the same, in which case the first N(NCODA + 1) elements of A will be destroyed.

WK — Work array of length 3 N.

2.Informational error

Type Code

31 The iteration for an eigenvalue failed to converge. The best estimate will be returned.

Algorithm

Routine EVASB computes the largest or smallest eigenvalues of a real band symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. The rational QR algorithm with Newton corrections is used to compute the extreme eigenvalues of this tridiagonal matrix.

The reduction routine is based on the EISPACK routine BANDR; see Garbow et al. (1978). The QR routine is based on the EISPACK routine RATQR; see Smith et al. (1976).

Example

The following example is given in Gregory and Karney (1969, page 63). The smallest four eigenvalues of the matrix