### ML_DGEBAK – Backward transformation to ML_DGEBAL of left or right eigenvectors

Block SymbolLicensing group: MATRIX

Function Description

The output references ySCALE and yV are always set to the corresponding input references
uSCALE and uV. If HLD = on then nothing is computed otherwise the LAPACK function
DGEBAK is called internally:

DGEBAK(sJOB, sRSIDE, N, ilo, IHI, uSCALE, M, uV, LDV, info);

where parameters of DGEBAK are set in the following way:

- Integer input job is mapped to the string sJOB: $\left\{0,1\right\}\to \mathtt{\text{"N"}}$, $\left\{2\right\}\to \mathtt{\text{"P"}}$, $\left\{3\right\}\to \mathtt{\text{"S"}}$ and $\left\{4\right\}\to \mathtt{\text{"B"}}$.
- If $\mathtt{\text{RSIDE}}=\mathtt{\text{on}}$ then the string sRSIDE is set to "R" else it is set to "L".
- N is number of elements of the vector referenced by uSCALE.
- If the input $\mathtt{\text{ihi}}\ne 0$ then IHI is set to ihi else IHI is set to $\mathtt{\text{N}}-1$.
- M is number of columns of the matrix referenced by uV.
- LDV is the leading dimension of the matrix referenced by uV.
- info is return code from the function DGEBAK.

The error flag E is set to on if:

- the reference uSCALE or uV is not defined (i.e. input uSCALE or uV is not connected),
- the call of the function DGEBAK returns error using the function XERBLA, see the return code info and system log.

Emphasize that the indices ilo and ihi start from zero unlike FORTRAN version where they start from one. See LAPACK documentation [7] for more details.

Inputs

uSCALE | Input reference to vector SCALE | Reference |

uV | Reference to matrix of right or left eigenvectors to be transformed | Reference |

job | Type of backward transformation required $\downarrow $0 $\uparrow $4 $\odot $0.00E+00 | Long (I32) |

RSIDE | Operation is applied from right side | Bool |

ilo | Zero based low row and column index of working submatrix $\odot $0.00E+00 | Long(I32) |

ihi | Zero based high row and column index of working submatrix $\odot $0.00E+00 | Long(I32) |

HLD | Hold | Bool |

Outputs

ySCALE | Output reference to vector SCALE | Reference |

yV | Reference to matrix of transformed right or left eigenvectors | Reference |

E | Error indicator | Bool |

info | LAPACK function result info. If info = -i, the i=th argument had an illegal value | Long (I32) |

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