The LA_EIGENPROBLEM function uses the QR algorithm to compute all eigenvalues λ and eigenvectors *v* ≠ 0 of an *n*-by-*n* real nonsymmetric or complex non-Hermitian array *A*, for the eigenproblem *Av* = λ*v*. The routine can also compute the left eigenvectors *u* ≠ 0, which satisfy *u ^{H}*

*A*= λ

*u*.

^{H}LA_EIGENPROBLEM may also be used for the generalized eigenproblem:

*Av =* λ*Bv* and *u ^{H}*

*A =*λ

*u*

^{H}*B*

where *A* and *B* are square arrays, *v* are the right eigenvectors, and *u* are the left eigenvectors.

LA_EIGENPROBLEM is based on the following LAPACK routines:

Output Type |
Standard LAPACK Routine |
Generalized LAPACK Routine |

Float |
sgeevx |
sggevx |

Double |
dgeevx |
dggevx |

Complex |
cgeevx |
cggevx |

Double complex |
zgeevx |
zggevx |

## Examples

Find the eigenvalues and eigenvectors for an array using the following program:

`PRO ExLA_EIGENPROBLEM`

`; Create a random array:`

`n = 4`

`seed = 12321`

`array = RANDOMN(seed, n, n)`

`; Compute all eigenvalues and eigenvectors:`

`eigenvalues = LA_EIGENPROBLEM(array, $`

EIGENVECTORS = eigenvectors)

PRINT, 'LA_EIGENPROBLEM Eigenvalues:'

`PRINT, eigenvalues`

`; Check the results using the eigenvalue equation:`

`maxErr = 0d`

FOR i = 0, n - 1 DO BEGIN

`; A*z = lambda*z`

alhs = array ## eigenvectors[*,i]

arhs = eigenvalues[i]*eigenvectors[*,i]

maxErr = maxErr > MAX(ABS(alhs - arhs))

`ENDFOR`

PRINT, 'LA_EIGENPROBLEM Error:', maxErr

`; Now try the generalized eigenproblem:`

b = IDENTITY(n) + 0.01*RANDOMN(seed, n, n)

`eigenvalues = LA_EIGENPROBLEM(Array, B)`

PRINT, 'LA_EIGENPROBLEM Generalized Eigenvalues:'

`PRINT, EIGENVALUES`

`END`

ExLA_EIGENPROBLEM

When this program is compiled and run, IDL prints:

LA_EIGENPROBLEM Eigenvalues:

( -1.50248, 0.592900)( -1.50248, -0.592900)( 1.99779, 0.000000)( 1.06499, 0.000000)

LA_EIGENPROBLEM Error: 9.5367432e-007

LA_EIGENPROBLEM Generalized Eigenvalues:

( -1.51732, 0.591994)( -1.51732, -0.591994)( 2.02658, 0.000000)( 1.04831, 0.000000)

## Syntax

*Result* = LA_EIGENPROBLEM( *A* [, *B*] [, ALPHA=*variable*] [, BALANCE=*value*] [, BETA=*variable*] [, /DOUBLE] [, EIGENVECTORS=*variable*] [, LEFT_EIGENVECTORS=*variable*] [, NORM_BALANCE =* variable*] [, PERMUTE_RESULT=*variable*] [, SCALE_RESULT=*variable*] [, RCOND_VALUE=*variable*] [, RCOND_VECTOR=*variable*] [, STATUS=*variable*] )

## Return Value

The result is a complex *n*-element vector containing the eigenvalues.

## Arguments

### A

The real or complex array for which to compute eigenvalues and eigenvectors.

### B

An optional real or complex* n*-by-*n *array used for the generalized eigenproblem. The elements of *B* are converted to the same type as *A* before computation.

## Keywords

### ALPHA

For the generalized eigenproblem with the *B* argument, set this keyword to a named variable in which the numerator of the eigenvalues will be returned as a complex *n *-element vector. For the standard eigenproblem this keyword is ignored.

*Tip: *The ALPHA and BETA values are useful for eigenvalues which underflow or overflow. In this case the eigenvalue problem may be rewritten as α*Av* = β*Bv*.

### BALANCE

Set this keyword to one of the following values:

- BALANCE = 0: No balancing is applied to
*A*. - BALANCE = 1: Both permutation and scale balancing are performed.
- BALANCE = 2: Permutations are performed to make the array more nearly upper triangular.
- BALANCE = 3: Diagonally scale the array to make the columns and rows more equal in norm.

The default is BALANCE = 1, which performs both permutation and scaling balances. Balancing a nonsymmetric (or non-Hermitian) array is recommended to reduce the sensitivity of eigenvalues to rounding errors.

### BETA

For the generalized eigenproblem with the *B* argument, set this keyword to a named variable in which the denominator of the eigenvalues will be returned as a real or complex *n*-element vector. For the standard eigenproblem this keyword is ignored.

*Tip: *The ALPHA and BETA values are useful for eigenvalues which underflow or overflow. In this case, the eigenvalue problem may be rewritten as α*Av* = β*Bv*.

### DOUBLE

Set this keyword to use double-precision for computations and to return a double-precision (real or complex) result. Set DOUBLE = 0 to use single-precision for computations and to return a single-precision (real or complex) result. The default is /DOUBLE if *A* is double precision, otherwise the default is DOUBLE = 0.

### EIGENVECTORS

Set this keyword to a named variable in which the eigenvectors will be returned as a set of row vectors. If this variable is omitted then eigenvectors will not be computed unless the RCOND_VALUE or RCOND_VECTOR keywords are present.

*Note: *For the standard eigenproblem the eigenvectors are normalized and rotated to have norm 1 and largest component real. For the generalized eigenproblem the eigenvectors are normalized so that the largest component has abs(real) + abs(imaginary) = 1.

### LEFT_EIGENVECTORS

Set this keyword to a named variable in which the left eigenvectors will be returned as a set of row vectors. If this variable is omitted then left eigenvectors will not be computed unless the RCOND_VALUE or RCOND_VECTOR keywords are present.

*Note: *Note - For the standard eigenproblem the eigenvectors are normalized and rotated to have norm 1 and largest component real. For the generalized eigenproblem the eigenvectors are normalized so that the largest component has abs(real) + abs(imaginary) = 1.

### NORM_BALANCE

Set this keyword to a named variable in which the one-norm of the balanced matrix will be returned. The one-norm is defined as the maximum value of the sum of absolute values of the columns. For the standard eigenproblem, this will be returned as a scalar value; for the generalized eigenproblem this will be returned as a two-element vector containing the *A* and *B* norms.

### PERMUTE_RESULT

Set this keyword to a named variable in which the result for permutation balancing will be returned as a two-element vector [*ilo*, *ihi*]. If permute balancing is not done then the values will be *ilo* = *1* and *ihi* = *n*.

### RCOND_VALUE

Set this keyword to a named variable in which the reciprocal condition numbers for the eigenvalues will be returned as an *n*-element vector. If RCOND_VALUE is present then left and right eigenvectors must be computed.

### RCOND_VECTOR

Set this keyword to a named variable in which the reciprocal condition numbers for the eigenvectors will be returned as an *n*-element vector. If RCOND_VECTOR is present then left and right eigenvectors must be computed.

### SCALE_RESULT

Set this keyword to a named variable in which the results for permute and scale balancing will be returned. For the standard eigenproblem, this will be returned as an *n*-element vector. For the generalized eigenproblem, this will be returned as a *n*-by-2 array with the first row containing the permute and scale factors for the left side of *A* and *B* and the second row containing the factors for the right side of *A* and *B*.

### STATUS

Set this keyword to a named variable that will contain the status of the computation. Possible values are:

- STATUS = 0: The computation was successful.
- STATUS > 0: The QR algorithm failed to compute all eigenvalues; no eigenvectors or condition numbers were computed. The STATUS value indicates that eigenvalues
*ilo:STATUS*(starting at index 1) did not converge; all other eigenvalues converged.

*Note: *If STATUS is not specified, any error messages will be output to the screen.

## Version History

5.6 |
Introduced |

## Resources and References

For details see Anderson et al., *LAPACK Users' Guide*, 3rd ed., SIAM, 1999.