# NAG CL Interfacef08yec (dtgsja)

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## 1Purpose

f08yec computes the generalized singular value decomposition (GSVD) of two real upper trapezoidal matrices $A$ and $B$, where $A$ is an $m×n$ matrix and $B$ is a $p×n$ matrix.
$A$ and $B$ are assumed to be in the form returned by f08vgc.

## 2Specification

 #include
 void f08yec (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer p, Integer n, Integer k, Integer l, double a[], Integer pda, double b[], Integer pdb, double tola, double tolb, double alpha[], double beta[], double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, Integer *ncycle, NagError *fail)
The function may be called by the names: f08yec, nag_lapackeig_dtgsja or nag_dtgsja.

## 3Description

f08yec computes the GSVD of the matrices $A$ and $B$ which are assumed to have the form as returned by f08vgc
 $A= { n-k-lklk0A12A13l00A23m-k-l000() , if ​ m-k-l ≥ 0; n-k-lklk0A12A13m-k00A23() , if ​ m-k-l < 0 ; B= n-k-lkll00B13p-l000() ,$
where the $k×k$ matrix ${A}_{12}$ and the $l×l$ matrix ${B}_{13}$ are nonsingular upper triangular, ${A}_{23}$ is $l×l$ upper triangular if $m-k-l\ge 0$ and is $\left(m-k\right)×l$ upper trapezoidal otherwise.
f08yec computes orthogonal matrices $Q$, $U$ and $V$, diagonal matrices ${D}_{1}$ and ${D}_{2}$, and an upper triangular matrix $R$ such that
 $UTAQ = D1 ( 0 R ) , VTBQ = D2 ( 0 R ) .$
Optionally $Q$, $U$ and $V$ may or may not be computed, or they may be premultiplied by matrices ${Q}_{1}$, ${U}_{1}$ and ${V}_{1}$ respectively.
If $\left(m-k-l\right)\ge 0$ then ${D}_{1}$, ${D}_{2}$ and $R$ have the form
 $D1= klkI0l0Cm-k-l00() ,$
 $D2= kll0Sp-l00() ,$
 $R = klkR11R12l0R22() ,$
where $C=\mathrm{diag}\left({\alpha }_{k+1},,,\dots ,,,{\alpha }_{k+l}\right)\text{, }S=\mathrm{diag}\left({\beta }_{k+1},,,\dots ,,,{\beta }_{k+l}\right)$.
If $\left(m-k-l\right)<0$ then ${D}_{1}$, ${D}_{2}$ and $R$ have the form
 $D1= km-kk+l-mkI00m-k0C0() ,$
 $D2= km-kk+l-mm-k0S0k+l-m00Ip-l000() ,$
 $R = km-kk+l-mkR11R12R13m-k0R22R23k+l-m00R33() ,$
where $C=\mathrm{diag}\left({\alpha }_{k+1},,,\dots ,,,{\alpha }_{m}\right)\text{, }S=\mathrm{diag}\left({\beta }_{k+1},,,\dots ,,,{\beta }_{m}\right)$.
In both cases the diagonal matrix $C$ has non-negative diagonal elements, the diagonal matrix $S$ has positive diagonal elements, so that $S$ is nonsingular, and ${C}^{2}+{S}^{2}=1$. See Section 2.3.5.3 of Anderson et al. (1999) for further information.
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
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: $\mathbf{jobu}$Nag_ComputeUType Input
On entry: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u must contain an orthogonal matrix ${U}_{1}$ on entry, and the product ${U}_{1}U$ is returned.
If ${\mathbf{jobu}}=\mathrm{Nag_InitU}$, u is initialized to the unit matrix, and the orthogonal matrix $U$ is returned.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, $\mathrm{Nag_InitU}$ or $\mathrm{Nag_NotU}$.
3: $\mathbf{jobv}$Nag_ComputeVType Input
On entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v must contain an orthogonal matrix ${V}_{1}$ on entry, and the product ${V}_{1}V$ is returned.
If ${\mathbf{jobv}}=\mathrm{Nag_InitV}$, v is initialized to the unit matrix, and the orthogonal matrix $V$ is returned.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, $\mathrm{Nag_InitV}$ or $\mathrm{Nag_NotV}$.
4: $\mathbf{jobq}$Nag_ComputeQType Input
On entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q must contain an orthogonal matrix ${Q}_{1}$ on entry, and the product ${Q}_{1}Q$ is returned.
If ${\mathbf{jobq}}=\mathrm{Nag_InitQ}$, q is initialized to the unit matrix, and the orthogonal matrix $Q$ is returned.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, $\mathrm{Nag_InitQ}$ or $\mathrm{Nag_NotQ}$.
5: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
6: $\mathbf{p}$Integer Input
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
7: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
8: $\mathbf{k}$Integer Input
9: $\mathbf{l}$Integer Input
On entry: k and l specify the sizes, $k$ and $l$, of the subblocks of $A$ and $B$, whose GSVD is to be computed by f08yec.
10: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m×n$ matrix $A$.
On exit: if $m-k-l\ge 0$, ${\mathbf{A}}\left(1:k+l,n-k-l+1:n\right)$ contains the $\left(k+l\right)×\left(k+l\right)$ upper triangular matrix $R$.
If $m-k-l<0$, ${\mathbf{A}}\left(1:m,n-k-l+1:n\right)$ contains the first $m$ rows of the $\left(k+l\right)×\left(k+l\right)$ upper triangular matrix $R$, and the submatrix ${R}_{33}$ is returned in ${\mathbf{B}}\left(m-k+1:l,n+m-k-l+1:n\right)$.
11: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{b}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
where ${\mathbf{B}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $p×n$ matrix $B$.
On exit: if $m-k-l<0$, ${\mathbf{B}}\left(m-k+1:l,n+m-k-l+1:n\right)$ contains the submatrix ${R}_{33}$ of $R$.
13: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14: $\mathbf{tola}$double Input
15: $\mathbf{tolb}$double Input
On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by f08vgc, say
 $tola=max(m,n)‖A‖ε, tolb=max(p,n)‖B‖ε,$
where $\epsilon$ is the machine precision.
16: $\mathbf{alpha}\left[{\mathbf{n}}\right]$double Output
On exit: see the description of beta.
17: $\mathbf{beta}\left[{\mathbf{n}}\right]$double Output
On exit: alpha and beta contain the generalized singular value pairs of $A$ and $B$;
• ${\mathbf{alpha}}\left[\mathit{i}\right]=1$, ${\mathbf{beta}}\left[\mathit{i}\right]=0$, for $\mathit{i}=0,1,\dots ,k-1$, and
• if $m-k-l\ge 0$, ${\mathbf{alpha}}\left[\mathit{i}\right]={\alpha }_{\mathit{i}}$, ${\mathbf{beta}}\left[\mathit{i}\right]={\beta }_{\mathit{i}}$, for $\mathit{i}=k,\dots ,k+l-1$, or
• if $m-k-l<0$, ${\mathbf{alpha}}\left[\mathit{i}\right]={\alpha }_{\mathit{i}}$, ${\mathbf{beta}}\left[\mathit{i}\right]={\beta }_{\mathit{i}}$, for $\mathit{i}=k,\dots ,m-1$ and ${\mathbf{alpha}}\left[\mathit{i}\right]=0$, ${\mathbf{beta}}\left[\mathit{i}\right]=1$, for $\mathit{i}=m,\dots ,k+l-1$.
Furthermore, if $k+l, ${\mathbf{alpha}}\left[\mathit{i}\right]={\mathbf{beta}}\left[\mathit{i}\right]=0$, for $\mathit{i}=k+l,\dots ,n-1$.
18: $\mathbf{u}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{m}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $U$ is stored in
• ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{pdu}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{u}}\left[\left(i-1\right)×{\mathbf{pdu}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u must contain an $m×m$ matrix ${U}_{1}$ (usually the orthogonal matrix returned by f08vgc).
On exit: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u contains the product ${U}_{1}U$.
If ${\mathbf{jobu}}=\mathrm{Nag_InitU}$, u contains the orthogonal matrix $U$.
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$, u is not referenced.
19: $\mathbf{pdu}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
20: $\mathbf{v}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array v must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdv}}×{\mathbf{p}}\right)$ when ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $V$ is stored in
• ${\mathbf{v}}\left[\left(j-1\right)×{\mathbf{pdv}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{v}}\left[\left(i-1\right)×{\mathbf{pdv}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v must contain an $p×p$ matrix ${V}_{1}$ (usually the orthogonal matrix returned by f08vgc).
On exit: if ${\mathbf{jobv}}=\mathrm{Nag_InitV}$, v contains the orthogonal matrix $V$.
If ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$, v contains the product ${V}_{1}V$.
If ${\mathbf{jobv}}=\mathrm{Nag_NotV}$, v is not referenced.
21: $\mathbf{pdv}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
• if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{pdv}}\ge 1$.
22: $\mathbf{q}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q must contain an $n×n$ matrix ${Q}_{1}$ (usually the orthogonal matrix returned by f08vgc).
On exit: if ${\mathbf{jobq}}=\mathrm{Nag_InitQ}$, q contains the orthogonal matrix $Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$, q contains the product ${Q}_{1}Q$.
If ${\mathbf{jobq}}=\mathrm{Nag_NotQ}$, q is not referenced.
23: $\mathbf{pdq}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$.
24: $\mathbf{ncycle}$Integer * Output
On exit: the number of cycles required for convergence.
25: $\mathbf{fail}$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_CONVERGENCE
The procedure does not converge after $40$ cycles.
NE_ENUM_INT_2
On entry, ${\mathbf{jobq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobq}}=\mathrm{Nag_ComputeQ}$ or $\mathrm{Nag_InitQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{jobu}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$ or $\mathrm{Nag_InitU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobv}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobv}}=\mathrm{Nag_ComputeV}$ or $\mathrm{Nag_InitV}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
otherwise ${\mathbf{pdv}}\ge 1$.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdv}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdv}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
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: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
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

The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $‖E‖2 = O⁡ε ‖A‖2 and ‖F‖2= O⁡ε ‖B‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08yec 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 complex analogue of this function is f08ysc.

## 10Example

This example finds the generalized singular value decomposition
 $A = UΣ1 ( 0 R ) QT , B= VΣ2 ( 0 R ) QT ,$
of the matrix pair $\left(A,B\right)$, where
 $A = ( 1 2 3 3 2 1 4 5 6 7 8 8 ) and B= ( −2 −3 3 4 6 5 ) .$

### 10.1Program Text

Program Text (f08yece.c)

### 10.2Program Data

Program Data (f08yece.d)

### 10.3Program Results

Program Results (f08yece.r)