# NAG CL Interfacef08sec (dsygst)

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## 1Purpose

f08sec reduces a real symmetric-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, where $A$ is a real symmetric matrix and $B$ has been factorized by f07fdc.

## 2Specification

 #include
 void f08sec (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, double a[], Integer pda, const double b[], Integer pdb, NagError *fail)
The function may be called by the names: f08sec, nag_lapackeig_dsygst or nag_dsygst.

## 3Description

To reduce the real symmetric-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, f08sec must be preceded by a call to f07fdc which computes the Cholesky factorization of $B$; $B$ must be positive definite.
The different problem types are specified by the argument comp_type, as indicated in the table below. The table shows how $C$ is computed by the function, and also how the eigenvectors $z$ of the original problem can be recovered from the eigenvectors of the standard form.
${\mathbf{order}}=\mathrm{Nag_ColMajor}$ ${\mathbf{order}}=\mathrm{Nag_RowMajor}$
comp_type Problem uplo $\mathbit{B}$ $\mathbit{C}$ $\mathbit{z}$ $\mathbit{B}$ $\mathbit{C}$ $\mathbit{z}$
$1$ $Az=\lambda Bz$ $\mathrm{Nag_Upper}$
$\mathrm{Nag_Lower}$
${U}^{\mathrm{T}}U$
$L{L}^{\mathrm{T}}$
${U}^{-\mathrm{T}}A{U}^{-1}$
${L}^{-1}A{L}^{-\mathrm{T}}$
${U}^{-1}y$
${L}^{-\mathrm{T}}y$
$U{U}^{\mathrm{T}}$
${L}^{\mathrm{T}}L$
${U}^{-1}A{U}^{-\mathrm{T}}$
${L}^{-\mathrm{T}}A{L}^{-1}$
${U}^{-\mathrm{T}}y$
${L}^{-1}y$
$2$ $ABz=\lambda z$ $\mathrm{Nag_Upper}$
$\mathrm{Nag_Lower}$
${U}^{\mathrm{T}}U$
$L{L}^{\mathrm{T}}$
$UA{U}^{\mathrm{T}}$
${L}^{\mathrm{T}}AL$
${U}^{-1}y$
${L}^{-\mathrm{T}}y$
$U{U}^{\mathrm{T}}$
${L}^{\mathrm{T}}L$
${U}^{\mathrm{T}}AU$
$LA{L}^{\mathrm{T}}$
${U}^{-\mathrm{T}}y$
${L}^{-1}y$
$3$ $BAz=\lambda z$ $\mathrm{Nag_Upper}$
$\mathrm{Nag_Lower}$
${U}^{\mathrm{T}}U$
$L{L}^{\mathrm{T}}$
$UA{U}^{\mathrm{T}}$
${L}^{\mathrm{T}}AL$
${U}^{\mathrm{T}}y$
$Ly$
$U{U}^{\mathrm{T}}$
${L}^{\mathrm{T}}L$
${U}^{\mathrm{T}}AU$
$LA{L}^{\mathrm{T}}$
$Uy$
${L}^{\mathrm{T}}y$

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{comp_type}$Nag_ComputeType Input
On entry: indicates how the standard form is computed.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C={U}^{-\mathrm{T}}A{U}^{-1}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={U}^{-1}A{U}^{-\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{-1}A{L}^{-\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={L}^{-\mathrm{T}}A{L}^{-1}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
${\mathbf{comp_type}}=\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $C=UA{U}^{\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C={U}^{\mathrm{T}}AU$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $C={L}^{\mathrm{T}}AL$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $C=LA{L}^{\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Constraint: ${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$, $\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$.
3: $\mathbf{uplo}$Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored and $B={U}^{\mathrm{T}}U$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $B=U{U}^{\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{T}}$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and $B={L}^{\mathrm{T}}L$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: the $n×n$ symmetric matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of a is overwritten by the corresponding upper or lower triangle of $C$ as specified by comp_type and uplo.
6: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7: $\mathbf{b}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $B$ as specified by uplo and returned by f07fdc.
8: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $B$ in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

Forming the reduced matrix $C$ is a stable procedure. However it involves implicit multiplication by ${B}^{-1}$ (if ${\mathbf{comp_type}}=\mathrm{Nag_Compute_1}$) or $B$ (if ${\mathbf{comp_type}}=\mathrm{Nag_Compute_2}$ or $\mathrm{Nag_Compute_3}$). When f08sec is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if $B$ is ill-conditioned with respect to inversion.

## 8Parallelism and Performance

f08sec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately ${n}^{3}$.
The complex analogue of this function is f08ssc.

## 10Example

This example computes all the eigenvalues of $Az=\lambda Bz$, where
 $A = ( 0.24 0.39 0.42 -0.16 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 -0.16 0.63 0.48 -0.03 ) and B= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.09 0.56 -0.83 0.76 0.34 -0.10 1.09 0.34 1.18 ) .$
Here $B$ is symmetric positive definite and must first be factorized by f07fdc. The program calls f08sec to reduce the problem to the standard form $Cy=\lambda y$; then f08fec to reduce $C$ to tridiagonal form, and f08jfc to compute the eigenvalues.

### 10.1Program Text

Program Text (f08sece.c)

### 10.2Program Data

Program Data (f08sece.d)

### 10.3Program Results

Program Results (f08sece.r)