Parameters

balanc

Type: System..::..String

On entry: specifies the balance option to be performed.

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

Do not diagonally scale or permute.

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

Permute only.

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

Scale only.

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

Both permute and scale.

Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, $\mathbf{balanc}=\text{"B"}$ is recommended.

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

jobvl

Type: System..::..String

On entry: if $\mathbf{jobvl}=\text{"N"}$, do not compute the left generalized eigenvectors.

If $\mathbf{jobvl}=\text{"V"}$, compute the left generalized eigenvectors.

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

jobvr

Type: System..::..String

On entry: if $\mathbf{jobvr}=\text{"N"}$, do not compute the right generalized eigenvectors.

If $\mathbf{jobvr}=\text{"V"}$, compute the right generalized eigenvectors.

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 eigenvectors only.

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

Computed for eigenvalues and eigenvectors.

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

n

Type: System..::..Int32

On entry: $n$, the order of the matrices $A$ and $B$.

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 matrix $A$ in the pair $\left(A,B\right)$.

On exit: a has been overwritten. If $\mathbf{jobvl}=\text{"V"}$ or $\mathbf{jobvr}=\text{"V"}$ or both, then $A$ contains the first part of the real Schur form of the ‘balanced’ versions of the input $A$ and $B$.

b

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 b must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the matrix $B$ in the pair $\left(A,B\right)$.

On exit: b has been overwritten.

alphar

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

An array of size [n]

On exit: the element $\mathbf{alphar}\left[j-1\right]$ contains the real part of ${\alpha }_j$.

alphai

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

An array of size [n]

On exit: the element $\mathbf{alphai}\left[j-1\right]$ contains the imaginary part of ${\alpha }_j$.

beta

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

An array of size [n]

On exit: $\left(\mathbf{alphar}\left[\mathit{j}-1\right]+\mathbf{alphai}\left[\mathit{j}-1\right]\times i\right)/\mathbf{beta}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$, will be the generalized eigenvalues.

If $\mathbf{alphai}\left[j-1\right]$ is zero, then the $j$th eigenvalue is real; if positive, then the $j$th and $\left(j+1\right)$st eigenvalues are a complex conjugate pair, with $\mathbf{alphai}\left[j\right]$ negative.

Note:  the quotients $\mathbf{alphar}\left[j-1\right]/\mathbf{beta}\left[j-1\right]$ and $\mathbf{alphai}\left[j-1\right]/\mathbf{beta}\left[j-1\right]$ may easily overflow or underflow, and $\mathbf{beta}\left[j-1\right]$ may even be zero. Thus, you should avoid naively computing the ratio ${\alpha }_j/{\beta }_j$. However, $\max \left|{\alpha }_j\right|$ will always be less than and usually comparable with ${‖\mathbf{a}‖}_2$ in magnitude, and $\max \left|{\beta }_j\right|$ will always be less than and usually comparable with ${‖\mathbf{b}‖}_2$.

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 generalized eigenvectors $u_j$ are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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 generalized eigenvectors $v_j$ are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have $\left|\text{real part}\right|+\left|\text{imag. part}\right|=1$.

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

ilo

Type: System..::..Int32%

On exit: ilo and ihi are integer values such that $\mathbf{a}[i-1,j-1]=0$ and $\mathbf{b}[i-1,j-1]=0$ if $i>j$ and $j=1,2,\dots ,\mathbf{ilo}-1$ or $i=\mathbf{ihi}+1,\dots ,\mathbf{n}$.

If $\mathbf{balanc}=\text{"N"}$ or $\text{"S"}$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=\mathbf{n}$.

ihi

Type: System..::..Int32%

On exit: ilo and ihi are integer values such that $\mathbf{a}[i-1,j-1]=0$ and $\mathbf{b}[i-1,j-1]=0$ if $i>j$ and $j=1,2,\dots ,\mathbf{ilo}-1$ or $i=\mathbf{ihi}+1,\dots ,\mathbf{n}$.

If $\mathbf{balanc}=\text{"N"}$ or $\text{"S"}$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=\mathbf{n}$.

lscale

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

An array of size [n]

On exit: details of the permutations and scaling factors applied to the left side of $A$ and $B$.

If ${\mathit{pl}}_j$ is the index of the row interchanged with row $j$, and ${\mathit{dl}}_j$ is the scaling factor applied to row $j$, then:

• $\mathbf{lscale}\left[\mathit{j}-1\right]={\mathit{pl}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{ilo}-1$;
• $\mathbf{lscale}={\mathit{dl}}_{\mathit{j}}$, for $\mathit{j}=\mathbf{ilo},\dots ,\mathbf{ihi}$;
• $\mathbf{lscale}={\mathit{pl}}_{\mathit{j}}$, 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 $\mathbf{ilo}-1$.

rscale

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

An array of size [n]

On exit: details of the permutations and scaling factors applied to the right side of $A$ and $B$.

If ${\mathit{pr}}_j$ is the index of the column interchanged with column $j$, and ${\mathit{dr}}_j$ is the scaling factor applied to column $j$, then:

• $\mathbf{rscale}\left[\mathit{j}-1\right]={\mathit{pr}}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{ilo}-1$;
• if $\mathbf{rscale}={\mathit{dr}}_{\mathit{j}}$, for $\mathit{j}=\mathbf{ilo},\dots ,\mathbf{ihi}$;
• if $\mathbf{rscale}={\mathit{pr}}_{\mathit{j}}$, 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 $A$.

bbnrm

Type: System..::..Double%

On exit: the $1$-norm of the balanced matrix $B$.

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: if $\mathbf{sense}=\text{"E"}$ or $\text{"B"}$, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus $\mathbf{rconde}\left[j-1\right]$, $\mathbf{rcondv}\left[j-1\right]$, and the $j$th columns of vl and vr all correspond to the $j$th eigenpair.

If $\mathbf{sense}=\text{"V"}$, rconde is not referenced.

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: if $\mathbf{sense}=\text{"V"}$ or $\text{"B"}$, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.

If $\mathbf{sense}=\text{"E"}$, rcondv is not referenced.

info

Type: System..::..Int32%

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