# NAG CL Interfacef08hec (dsbtrd)

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## 1Purpose

f08hec reduces a real symmetric band matrix to tridiagonal form.

## 2Specification

 #include
 void f08hec (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double d[], double e[], double q[], Integer pdq, NagError *fail)
The function may be called by the names: f08hec, nag_lapackeig_dsbtrd or nag_dsbtrd.

## 3Description

f08hec reduces a symmetric band matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation:
 $T = QT A Q .$
The orthogonal matrix $Q$ is determined as a product of Givens rotation matrices, and may be formed explicitly by the function if required.
The function uses a vectorizable form of the reduction, due to Kaufman (1984).

## 4References

Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

## 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: $\mathbf{vect}$Nag_VectType Input
On entry: indicates whether $Q$ is to be returned.
${\mathbf{vect}}=\mathrm{Nag_FormQ}$
$Q$ is returned.
${\mathbf{vect}}=\mathrm{Nag_UpdateQ}$
$Q$ is updated (and the array q must contain a matrix on entry).
${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$
$Q$ is not required.
Constraint: ${\mathbf{vect}}=\mathrm{Nag_FormQ}$, $\mathrm{Nag_UpdateQ}$ or $\mathrm{Nag_DoNotForm}$.
3: $\mathbf{uplo}$Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{kd}$Integer Input
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the number of superdiagonals, ${k}_{d}$, of the matrix $A$.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the number of subdiagonals, ${k}_{d}$, of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
6: $\mathbf{ab}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the upper or lower triangle of the $n×n$ symmetric band matrix $A$.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of ${A}_{ij}$, depends on the order and uplo arguments as follows:
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{d}+i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right),\dots ,j$;
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{d}\right)$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{d}+j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{d}\right),\dots ,i$.
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in ab using the same storage format as described above.
7: $\mathbf{pdab}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{kd}}+1\right)$.
8: $\mathbf{d}\left[{\mathbf{n}}\right]$double Output
On exit: the diagonal elements of the tridiagonal matrix $T$.
9: $\mathbf{e}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On exit: the off-diagonal elements of the tridiagonal matrix $T$.
10: $\mathbf{q}\left[\mathit{dim}\right]$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 ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ or $\mathrm{Nag_UpdateQ}$;
• $1$ when ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$.
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 ${\mathbf{vect}}=\mathrm{Nag_UpdateQ}$, q must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded symmetric-definite generalized eigenproblem); otherwise q need not be set.
On exit: if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ or $\mathrm{Nag_UpdateQ}$, the $n×n$ matrix $Q$.
If ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, q is not referenced.
11: $\mathbf{pdq}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ or $\mathrm{Nag_UpdateQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, ${\mathbf{pdq}}\ge 1$.
12: $\mathbf{fail}$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_ENUM_INT_2
On entry, ${\mathbf{vect}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ or $\mathrm{Nag_UpdateQ}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, ${\mathbf{pdq}}\ge 1$.
NE_INT
On entry, ${\mathbf{kd}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{kd}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}>0$.
On entry, ${\mathbf{pdq}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdq}}>0$.
NE_INT_2
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{kd}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{kd}}+1\right)$.
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 tridiagonal matrix $T$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $‖E‖2≤ c (n) ε ‖A‖2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ is the machine precision.
The elements of $T$ themselves may be sensitive to small perturbations in $A$ or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.
The computed matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $‖E‖2 = O(ε) ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08hec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hec 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 $6{n}^{2}k$ if ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$ with $3{n}^{3}\left(k-1\right)/k$ additional operations if ${\mathbf{vect}}=\mathrm{Nag_FormQ}$.
The complex analogue of this function is f08hsc.

## 10Example

This example computes all the eigenvalues and eigenvectors of the matrix $A$, where
 $A = ( 4.99 0.04 0.22 0.00 0.04 1.05 -0.79 1.04 0.22 -0.79 -2.31 -1.30 0.00 1.04 -1.30 -0.43 ) .$
Here $A$ is symmetric and is treated as a band matrix. The program first calls f08hec to reduce $A$ to tridiagonal form $T$, and to form the orthogonal matrix $Q$; the results are then passed to f08jec which computes the eigenvalues and eigenvectors of $A$.

### 10.1Program Text

Program Text (f08hece.c)

### 10.2Program Data

Program Data (f08hece.d)

### 10.3Program Results

Program Results (f08hece.r)