### 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  $↓$0 $↑$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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