Parameters

balanc

Type: System..::..String

On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.

$\mathbf{balanc}=\text{"N"}$

Do not diagonally scale or permute.

$\mathbf{balanc}=\text{"P"}$

Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.

$\mathbf{balanc}=\text{"S"}$

Diagonally scale the matrix, i.e., replace $A$ by $DA{D}^{-1}$, where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.

$\mathbf{balanc}=\text{"B"}$

Both diagonally scale and permute $A$.

Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

Constraint: $\mathbf{balanc}=\text{"N"}$, $\text{"P"}$, $\text{"S"}$ or $\text{"B"}$.

jobvl

Type: System..::..String

On entry: if $\mathbf{jobvl}=\text{"N"}$, the left eigenvectors of $A$ are not computed.

If $\mathbf{jobvl}=\text{"V"}$, the left eigenvectors of $A$ are computed.

If $\mathbf{sense}=\text{"E"}$ or $\text{"B"}$, jobvl must be set to $\mathbf{jobvl}=\text{"V"}$.

Constraint: $\mathbf{jobvl}=\text{"N"}$ or $\text{"V"}$.

jobvr

Type: System..::..String

On entry: if $\mathbf{jobvr}=\text{"N"}$, the right eigenvectors of $A$ are not computed.

If $\mathbf{jobvr}=\text{"V"}$, the right eigenvectors of $A$ are computed.

If $\mathbf{sense}=\text{"E"}$ or $\text{"B"}$, jobvr must be set to $\mathbf{jobvr}=\text{"V"}$.

Constraint: $\mathbf{jobvr}=\text{"N"}$ or $\text{"V"}$.

sense

Type: System..::..String

On entry: determines which reciprocal condition numbers are computed.

$\mathbf{sense}=\text{"N"}$

None are computed.

$\mathbf{sense}=\text{"E"}$

Computed for eigenvalues only.

$\mathbf{sense}=\text{"V"}$

Computed for right eigenvectors only.

$\mathbf{sense}=\text{"B"}$

Computed for eigenvalues and right eigenvectors.

If $\mathbf{sense}=\text{"E"}$ or $\text{"B"}$, both left and right eigenvectors must also be computed ($\mathbf{jobvl}=\text{"V"}$ and $\mathbf{jobvr}=\text{"V"}$).

Constraint: $\mathbf{sense}=\text{"N"}$, $\text{"E"}$, $\text{"V"}$ or $\text{"B"}$.

n

Type: System..::..Int32

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

a

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, dim2]

Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \max \left(1,\mathbf{n}\right)$

Note: the second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the $n$ by $n$ matrix $A$.

On exit: a has been overwritten. If $\mathbf{jobvl}=\text{"V"}$ or $\mathbf{jobvr}=\text{"V"}$, $A$ contains the real Schur form of the balanced version of the input matrix $A$.

wr

Type: array<System..::..Double>[]()[][]

An array of size [dim1]

Note: the dimension of the arrays wr and wi must be at least $\max \left(1,\mathbf{n}\right)$.

On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

wi

Type: array<System..::..Double>[]()[][]

An array of size [dim1]

Note: the dimension of the arrays wr and wi must be at least $\max \left(1,\mathbf{n}\right)$.

On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

vl

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, dim2]

Note: dim1 must satisfy the constraint:

• if $\mathbf{jobvl}=\text{"V"}$, $\mathrm{dim1}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathrm{dim1}\ge 1$.

Note: the second dimension of the array vl must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvl}=\text{"V"}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobvl}=\text{"V"}$, the left eigenvectors $u_j$ are stored one after another in the columns of vl, in the same order as their corresponding eigenvalues. If the $j$th eigenvalue is real, then $u_j=\mathbf{vl}[\mathit{i}-1,j-1]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $u_j=\mathbf{vl}[\mathit{i}-1,j-1]+\mathit{i}\times \mathbf{vl}[\mathit{i}-1,j]$ and $u_{j+1}=\mathbf{vl}[\mathit{i}-1,j-1]-\mathit{i}\times \mathbf{vl}[\mathit{i}-1,j]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{jobvl}=\text{"N"}$, vl is not referenced.

vr

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1, dim2]

Note: dim1 must satisfy the constraint:

• if $\mathbf{jobvr}=\text{"V"}$, $\mathrm{dim1}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathrm{dim1}\ge 1$.

Note: the second dimension of the array vr must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvr}=\text{"V"}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobvr}=\text{"V"}$, the right eigenvectors $v_j$ are stored one after another in the columns of vr, in the same order as their corresponding eigenvalues. If the $j$th eigenvalue is real, then $v_j=\mathbf{vr}[\mathit{i}-1,j-1]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $v_j=\mathbf{vr}[\mathit{i}-1,j-1]+\mathit{i}\times \mathbf{vr}[\mathit{i}-1,j]$ and $v_{j+1}=\mathbf{vr}[\mathit{i}-1,j-1]-\mathit{i}\times \mathbf{vr}[\mathit{i}-1,j]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{jobvr}=\text{"N"}$, vr is not referenced.

ilo

Type: System..::..Int32%

On exit: ilo and ihi are integer values determined when $A$ was balanced. The balanced $A$ has $a_{ij}=0$ if $i>j$ and $j=1,2,\dots ,\mathbf{ilo}-1$ or $i=\mathbf{ihi}+1,\dots ,\mathbf{n}$.

ihi

Type: System..::..Int32%

On exit: ilo and ihi are integer values determined when $A$ was balanced. The balanced $A$ has $a_{ij}=0$ if $i>j$ and $j=1,2,\dots ,\mathbf{ilo}-1$ or $i=\mathbf{ihi}+1,\dots ,\mathbf{n}$.

scale

Type: array<System..::..Double>[]()[][]

An array of size [dim1]

Note: the dimension of the array scale must be at least $\max \left(1,\mathbf{n}\right)$.

On exit: details of the permutations and scaling factors applied when balancing $A$.

If $p_j$ is the index of the row and column interchanged with row and column $j$, and $d_j$ is the scaling factor applied to row and column $j$, then

• $\mathbf{scale}\left[\mathit{j}-1\right]=p_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{ilo}-1$;
• $\mathbf{scale}\left[\mathit{j}-1\right]=d_{\mathit{j}}$, for $\mathit{j}=\mathbf{ilo},\dots ,\mathbf{ihi}$;
• $\mathbf{scale}\left[\mathit{j}-1\right]=p_{\mathit{j}}$, for $\mathit{j}=\mathbf{ihi}+1,\dots ,\mathbf{n}$.

The order in which the interchanges are made is n to $\mathbf{ihi}+1$, then $1$ to $\mathbf{ilo}-1$.

abnrm

Type: System..::..Double%

On exit: the $1$-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

rconde

Type: array<System..::..Double>[]()[][]

An array of size [dim1]

Note: the dimension of the array rconde must be at least $\max \left(1,\mathbf{n}\right)$.

On exit: $\mathbf{rconde}\left[j-1\right]$ is the reciprocal condition number of the $j$th eigenvalue.

rcondv

Type: array<System..::..Double>[]()[][]

An array of size [dim1]

Note: the dimension of the array rcondv must be at least $\max \left(1,\mathbf{n}\right)$.

On exit: $\mathbf{rcondv}\left[j-1\right]$ is the reciprocal condition number of the $j$th right eigenvector.

info

Type: System..::..Int32%

On exit: $\mathbf{info}=0$ unless the method detects an error (see [Error Indicators and Warnings]).