# NAG FL InterfaceF07 (Lapacklin)Linear Equations (LAPACK)

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## 1Scope of the Chapter

This chapter provides routines for the solution of systems of simultaneous linear equations, and associated computations. It provides routines for
• matrix factorizations;
• solution of linear equations;
• estimating matrix condition numbers;
• computing error bounds for the solution of linear equations;
• matrix inversion;
• computing scaling factors to equilibrate a matrix.
Routines are provided for both real and complex data.
For a general introduction to the solution of systems of linear equations, you should turn first to the F04 Chapter Introduction. The decision trees, in Section 4 in the F04 Chapter Introduction, direct you to the most appropriate routines in Chapters F04 or F07 for solving your particular problem. In particular, Chapters F04 and F07 contain Black Box (or driver) routines which enable some standard types of problem to be solved by a call to a single routine. Where possible, routines in Chapter F04 call Chapter F07 routines to perform the necessary computational tasks.
There are two types of driver routines in this chapter: simple drivers which just return the solution to the linear equations; and expert drivers which also return condition and error estimates and, in many cases, also allow equilibration. The simple drivers for real matrices have names of the form F07_AF (D__SV) and for complex matrices have names of the form F07_NF (Z__SV). The expert drivers for real matrices have names of the form F07_BF (D__SVX) and for complex matrices have names of the form F07_PF (Z__SVX).
The routines in this chapter (Chapter F07) handle only dense and band matrices (not matrices with more specialised structures, or general sparse matrices).
The routines in this chapter have all been derived from the LAPACK project (see Anderson et al. (1999)). They have been designed to be efficient on a wide range of high-performance computers, without compromising efficiency on conventional serial machines.

## 2Background to the Problems

This section is only a brief introduction to the numerical solution of systems of linear equations. Consult a standard textbook, for example Golub and Van Loan (1996) for a more thorough discussion.

### 2.1Notation

We use the standard notation for a system of simultaneous linear equations:
 $Ax=b$ (1)
where $A$ is the coefficient matrix, $b$ is the right-hand side, and $x$ is the solution. $A$ is assumed to be a square matrix of order $n$.
If there are several right-hand sides, we write
 $AX=B$ (2)
where the columns of $B$ are the individual right-hand sides, and the columns of $X$ are the corresponding solutions.
We also use the following notation, both here and in the routine documents:
 $\stackrel{^}{x}$ a computed solution to $Ax=b$, (which usually differs from the exact solution $x$ because of round-off error) $r=b-A\stackrel{^}{x}$ the residual corresponding to the computed solution $\stackrel{^}{x}$ ${‖x‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{x}_{i}|$ the $\infty$-norm of the vector $x$ ${‖x‖}_{1}=\sum _{j=1}^{n}|{x}_{j}|$ the $1$-norm of the vector $x$ ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ the $\infty$-norm of the matrix $A$ ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{n}|{a}_{ij}|$ the $1$-norm of the matrix $A$ $|x|$ the vector with elements $|{x}_{i}|$ $|A|$ the matrix with elements $|{a}_{ij}|$
Inequalities of the form $|A|\le |B|$ are interpreted component-wise, that is $|{a}_{ij}|\le |{b}_{ij}|$ for all $i,j$.

### 2.2Matrix Factorizations

If $A$ is upper or lower triangular, $Ax=b$ can be solved by a straightforward process of backward or forward substitution.
Otherwise, the solution is obtained after first factorizing $A$, as follows.
General matrices (LU factorization with partial pivoting)
 $A=PLU$
where $P$ is a permutation matrix, $L$ is lower-triangular with diagonal elements equal to $1$, and $U$ is upper-triangular; the permutation matrix $P$ (which represents row interchanges) is needed to ensure numerical stability.
Symmetric positive definite matrices (Cholesky factorization)
 $A=UTU or A=LLT$
where $U$ is upper triangular and $L$ is lower triangular.
Symmetric positive semidefinite matrices (pivoted Cholesky factorization)
 $A=PUTUPT or A=PLLTPT$
where $P$ is a permutation matrix, $U$ is upper triangular and $L$ is lower triangular. The permutation matrix $P$ (which represents row-and-column interchanges) is needed to ensure numerical stability and to reveal the numerical rank of $A$.
Symmetric indefinite matrices (Bunch–Kaufman factorization)
 $A = PUD UT PT or A = PLD LT PT$
where $P$ is a permutation matrix, $U$ is upper triangular, $L$ is lower triangular, and $D$ is a block diagonal matrix with diagonal blocks of order $1$ or $2$; $U$ and $L$ have diagonal elements equal to $1$, and have $2×2$ unit matrices on the diagonal corresponding to the $2×2$ blocks of $D$. The permutation matrix $P$ (which represents symmetric row-and-column interchanges) and the $2×2$ blocks in $D$ are needed to ensure numerical stability. If $A$ is in fact positive definite, no interchanges are needed and the factorization reduces to $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ with diagonal $D$, which is simply a variant form of the Cholesky factorization.

### 2.3Solution of Systems of Equations

Given one of the above matrix factorizations, it is straightforward to compute a solution to $Ax=b$ by solving two subproblems, as shown below, first for $y$ and then for $x$. Each subproblem consists essentially of solving a triangular system of equations by forward or backward substitution; the permutation matrix $P$ and the block diagonal matrix $D$ introduce only a little extra complication:
General matrices ( LU factorization)
 $Ly=PTb Ux=y$
Symmetric positive definite matrices (Cholesky factorization)
 $UTy=b Ux=y or Ly=b LTx=y$
Symmetric indefinite matrices (Bunch–Kaufman factorization)
 $PUDy=b UTPTx=y or PLDy=b LTPTx=y$

### 2.4Sensitivity and Error Analysis

#### 2.4.1Normwise error bounds

Frequently, in practical problems the data $A$ and $b$ are not known exactly, and it is then important to understand how uncertainties or perturbations in the data can affect the solution.
If $x$ is the exact solution to $Ax=b$, and $x+\delta x$ is the exact solution to a perturbed problem $\left(A+\delta A\right)\left(x+\delta x\right)=\left(b+\delta b\right)$, then
 $‖δx‖ ‖x‖ ≤κ(A) (‖δA‖ ‖A‖ +‖δb‖ ‖b‖ )+⋯(second-order terms)$
where $\kappa \left(A\right)$ is the condition number of $A$ defined by
 $κ(A) = ‖A‖.‖A-1‖ .$ (3)
In other words, relative errors in $A$ or $b$ may be amplified in $x$ by a factor $\kappa \left(A\right)$. Section 2.4.2 discusses how to compute or estimate $\kappa \left(A\right)$.
Similar considerations apply when we study the effects of rounding errors introduced by computation in finite precision. The effects of rounding errors can be shown to be equivalent to perturbations in the original data, such that $\frac{‖\delta A‖}{‖A‖}$ and $\frac{‖\delta b‖}{‖b‖}$ are usually at most $p\left(n\right)\epsilon$, where $\epsilon$ is the machine precision and $p\left(n\right)$ is an increasing function of $n$ which is seldom larger than $10n$ (although in theory it can be as large as ${2}^{n-1}$).
In other words, the computed solution $\stackrel{^}{x}$ is the exact solution of a linear system $\left(A+\delta A\right)\stackrel{^}{x}=b+\delta b$ which is close to the original system in a normwise sense.

#### 2.4.2Estimating condition numbers

The previous section has emphasized the usefulness of the quantity $\kappa \left(A\right)$ in understanding the sensitivity of the solution of $Ax=b$. To compute the value of $\kappa \left(A\right)$ from equation (3) is more expensive than solving $Ax=b$ in the first place. Hence it is standard practice to estimate $\kappa \left(A\right)$, in either the $1$-norm or the $\infty$-norm, by a method which only requires $\mathit{O}\left({n}^{2}\right)$ additional operations, assuming that a suitable factorization of $A$ is available.
The method used in this chapter is Higham's modification of Hager's method (see Higham (1988)). It yields an estimate which is never larger than the true value, but which seldom falls short by more than a factor of $3$ (although artificial examples can be constructed where it is much smaller). This is acceptable since it is the order of magnitude of $\kappa \left(A\right)$ which is important rather than its precise value.
Because $\kappa \left(A\right)$ is infinite if $A$ is singular, the routines in this chapter actually return the reciprocal of $\kappa \left(A\right)$.

#### 2.4.3Scaling and Equilibration

The condition of a matrix and hence the accuracy of the computed solution, may be improved by scaling; thus if ${D}_{1}$ and ${D}_{2}$ are diagonal matrices with positive diagonal elements, then
 $B = D1 A D2$
is the scaled matrix. A general matrix is said to be equilibrated if it is scaled so that the lengths of its rows and columns have approximately equal magnitude. Similarly a general matrix is said to be row-equilibrated (column-equilibrated) if it is scaled so that the lengths of its rows (columns) have approximately equal magnitude. Note that row scaling can affect the choice of pivot when partial pivoting is used in the factorization of $A$.
A symmetric or Hermitian positive definite matrix is said to be equilibrated if the diagonal elements are all approximately equal to unity.
For further information on scaling and equilibration see Section 3.5.2 of Golub and Van Loan (1996), Section 7.2, 7.3 and 9.8 of Higham (1988) and Section 5 of Chapter 4 of Wilkinson (1965).
Routines are provided to return the scaling factors that equilibrate a matrix for general, general band, symmetric and Hermitian positive definite and symmetric and Hermitian positive definite band matrices.

#### 2.4.4Componentwise error bounds

A disadvantage of normwise error bounds is that they do not reflect any special structure in the data $A$ and $b$ – that is, a pattern of elements which are known to be zero – and the bounds are dominated by the largest elements in the data.
Componentwise error bounds overcome these limitations. Instead of the normwise relative error, we can bound the relative error in each component of $A$ and $b$:
 $maxijk (|δaij| |aij| ,|δbk| |bk| ) ≤ω$
where the component-wise backward error bound $\omega$ is given by
 $ω= maxi |ri| (|A|.|x^|+|b|)i .$
Routines are provided in this chapter which compute $\omega$, and also compute a forward error bound which is sometimes much sharper than the normwise bound given earlier:
 $‖x-x^‖∞ ‖x‖∞ ≤ ‖|A-1|.|r|‖∞ ‖x‖∞ .$
Care is taken when computing this bound to allow for rounding errors in computing $r$. The norm ${‖|{A}^{-1}|.|r|‖}_{\infty }$ is estimated cheaply (without computing ${A}^{-1}$) by a modification of the method used to estimate $\kappa \left(A\right)$.

#### 2.4.5Iterative refinement of the solution

If $\stackrel{^}{x}$ is an approximate computed solution to $Ax=b$, and $r$ is the corresponding residual, then a procedure for iterative refinement of $\stackrel{^}{x}$ can be defined as follows, starting with ${x}_{0}=\stackrel{^}{x}$:
• for $i=0,1,\dots \text{}$, until convergence
 compute ${r}_{i}=b-A{x}_{i}$ solve $A{d}_{i}={r}_{i}$ compute ${x}_{i+1}={x}_{i}+{d}_{i}$
In Chapter F04, routines are provided which perform this procedure using additional precision to compute $r$, and are thus able to reduce the forward error to the level of machine precision.
The routines in this chapter do not use additional precision to compute $r$, and cannot guarantee a small forward error, but can guarantee a small backward error (except in rare cases when $A$ is very ill-conditioned, or when $A$ and $x$ are sparse in such a way that $|A|.|x|$ has a zero or very small component). The iterations continue until the backward error has been reduced as much as possible; usually only one iteration is needed.

### 2.5Matrix Inversion

It is seldom necessary to compute an explicit inverse of a matrix. In particular, do not attempt to solve $Ax=b$ by first computing ${A}^{-1}$ and then forming the matrix-vector product $x={A}^{-1}b$; the procedure described in Section 2.3 is more efficient and more accurate.
However, routines are provided for the rare occasions when an inverse is needed, using one of the factorizations described in Section 2.2.

### 2.6Packed Storage Formats

Routines which handle symmetric matrices are usually designed so that they use either the upper or lower triangle of the matrix; it is not necessary to store the whole matrix. If the upper or lower triangle is stored conventionally in the upper or lower triangle of a two-dimensional array, the remaining elements of the array can be used to store other useful data.
However, that is not always convenient, and if it is important to economize on storage, the upper or lower triangle can be stored in a one-dimensional array of length $n\left(n+1\right)/2$ or a two-dimensional array with $n\left(n+1\right)/2$ elements; in other words, the storage is almost halved.
The one-dimensional array storage format is referred to as packed storage; it is described in Section 3.3.2. The two-dimensional array storage format is referred to as Rectangular Full Packed (RFP) format; it is described in Section 3.3.3. They may also be used for triangular matrices.
Routines designed for these packed storage formats perform the same number of arithmetic operations as routines which use conventional storage. Those using a packed one-dimensional array are usually less efficient, especially on high-performance computers, so there is then a trade-off between storage and efficiency. The RFP routines are as efficient as for conventional storage, although only a small subset of routines use this format.

### 2.7Band and Tridiagonal Matrices

A band matrix is one whose nonzero elements are confined to a relatively small number of subdiagonals or superdiagonals on either side of the main diagonal. A tridiagonal matrix is a special case of a band matrix with just one subdiagonal and one superdiagonal. Algorithms can take advantage of bandedness to reduce the amount of work and storage required. The storage scheme used for band matrices is described in Section 3.3.4.
The $LU$ factorization for general matrices, and the Cholesky factorization for symmetric and Hermitian positive definite matrices both preserve bandedness. Hence routines are provided which take advantage of the band structure when solving systems of linear equations.
The Cholesky factorization preserves bandedness in a very precise sense: the factor $U$ or $L$ has the same number of superdiagonals or subdiagonals as the original matrix. In the $LU$ factorization, the row-interchanges modify the band structure: if $A$ has ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, then $L$ is not a band matrix but still has at most ${k}_{l}$ nonzero elements below the diagonal in each column; and $U$ has at most ${k}_{l}+{k}_{u}$ superdiagonals.
The Bunch–Kaufman factorization does not preserve bandedness, because of the need for symmetric row-and-column permutations; hence no routines are provided for symmetric indefinite band matrices.
The inverse of a band matrix does not in general have a band structure, so no routines are provided for computing inverses of band matrices.

### 2.8Block Partitioned Algorithms

Many of the routines in this chapter use what is termed a block partitioned algorithm. This means that at each major step of the algorithm a block of rows or columns is updated, and most of the computation is performed by matrix-matrix operations on these blocks. The matrix-matrix operations are performed by calls to the Level 3 BLAS (see Chapter F06), which are the key to achieving high performance on many modern computers. See Golub and Van Loan (1996) or Anderson et al. (1999) for more about block partitioned algorithms.
The performance of a block partitioned algorithm varies to some extent with the block size – that is, the number of rows or columns per block. This is a machine-dependent argument, which is set to a suitable value when the Library is implemented on each range of machines. You do not normally need to be aware of what value is being used. Different block sizes may be used for different routines. Values in the range $16$ to $64$ are typical.
On some machines there may be no advantage from using a block partitioned algorithm, and then the routines use an unblocked algorithm (effectively a block size of $1$), relying solely on calls to the Level 2 BLAS (see Chapter F06 again).
The only situation in which you need some awareness of the block size is when it affects the amount of workspace to be supplied to a particular routine. This is discussed in Section 3.4.3.

### 2.9Mixed Precision LAPACK Routines

Some LAPACK routines use mixed precision arithmetic in an effort to solve problems more efficiently on modern hardware. They work by converting a double precision problem into an equivalent single precision problem, solving it and then using iterative refinement in double precision to find a full precision solution to the original problem. The method may fail if the problem is too ill-conditioned to allow the initial single precision solution, in which case the routines fall back to solve the original problem entirely in double precision. The vast majority of problems are not so ill-conditioned, and in those cases the technique can lead to significant gains in speed without loss of accuracy. This is particularly true on machines where double precision arithmetic is significantly slower than single precision.

## 3Recommendations on Choice and Use of Available Routines

### 3.1Available Routines

Tables 1 to 8 in Section 3.5 show the routines which are provided for performing different computations on different types of matrices. Tables 1 to 4 show routines for real matrices; Tables 5 to 8 show routines for complex matrices. Each entry in the table gives the NAG routine name and the LAPACK double precision name (see Section 3.2).
Routines are provided for the following types of matrix:
• general
• general band
• general tridiagonal
• symmetric or Hermitian positive definite
• symmetric or Hermitian positive definite (packed storage)
• symmetric or Hermitian positive definite (RFP storage)
• symmetric or Hermitian positive definite band
• symmetric or Hermitian positive definite tridiagonal
• symmetric or Hermitian indefinite
• symmetric or Hermitian indefinite (packed storage)
• triangular
• triangular (packed storage)
• triangular (RFP storage)
• triangular band
For each of the above types of matrix (except where indicated), routines are provided to perform the following computations:
1. (a)(except for RFP matrices) solve a system of linear equations (driver routines);
2. (b)(except for RFP matrices) solve a system of linear equations with condition and error estimation (expert drivers);
3. (c)(except for triangular matrices) factorize the matrix (see Section 2.2);
4. (d)solve a system of linear equations, using the factorization (see Section 2.3);
5. (e)(except for RFP matrices) estimate the condition number of the matrix, using the factorization (see Section 2.4.2); these routines also require the norm of the original matrix (except when the matrix is triangular) which may be computed by a routine in Chapter F06;
6. (f)(except for RFP matrices) refine the solution and compute forward and backward error bounds (see Sections 2.4.4 and 2.4.5); these routines require the original matrix and right-hand side, as well as the factorization returned from (a) and the solution returned from (b);
7. (g)(except for band and tridiagonal matrices) invert the matrix, using the factorization (see Section 2.5);
8. (h)(except for tridiagonal, symmetric indefinite, triangular and RFP matrices) compute scale factors to equilibrate the matrix (see Section 2.4.3).
Thus, to solve a particular problem, it is usually only necessary to call a single driver routine, but alternatively two or more routines may be called in succession. This is illustrated in the example programs in the routine documents.

### 3.2NAG Names and LAPACK Names

As well as the NAG routine name (beginning F07), Tables 1 to 8 show the LAPACK routine names in double precision.
The routines may be called either by their NAG names or by their LAPACK names. When using the NAG Library, the double precision form of the LAPACK name must be used (beginning with D- or Z-).
References to Chapter F07 routines in the manual normally include the LAPACK double precision names, for example, f07adf.
The LAPACK routine names follow a simple scheme (which is similar to that used for the BLAS in Chapter F06). Most names have the structure XYYZZZ, where the components have the following meanings:
• the initial letter X indicates the data type (real or complex) and precision:  S real, single precision (in Fortran 77, REAL) D real, double precision (in Fortran 77, DOUBLE PRECISION) C complex, single precision (in Fortran 77, COMPLEX) Z complex, double precision (in Fortran 77, COMPLEX*16 or DOUBLE COMPLEX)
• exceptionally, the mixed precision LAPACK routines described in Section 2.9 replace the initial first letter by a pair of letters, as:  DS double precision routine using single precision internally ZC double complex routine using single precision complex internally
• the letters YY indicate the type of the matrix $A$ (and in some cases its storage scheme):  GE general GB general band PO symmetric or Hermitian positive definite PF symmetric or Hermitian positive definite (RFP storage) PP symmetric or Hermitian positive definite (packed storage) PB symmetric or Hermitian positive definite band SY symmetric indefinite SF symmetric indefinite (RFP storage) SP symmetric indefinite (packed storage) HE (complex) Hermitian indefinite HF (complex) Hermitian indefinite (RFP storage) HP (complex) Hermitian indefinite (packed storage) GT general tridiagonal PT symmetric or Hermitian positive definite tridiagonal TR triangular TF triangular (RFP storage) TP triangular (packed storage) TB triangular band
• the last two or three letters ZZ or ZZZ indicate the computation performed. Examples are:  TRF triangular factorization TRS solution of linear equations, using the factorization CON estimate condition number RFS refine solution and compute error bounds TRI compute inverse, using the factorization
Thus the routine dgetrf performs a triangular factorization of a real general matrix in double precision; the corresponding routine for a complex general matrix is zgetrf.

### 3.3Matrix Storage Schemes

In this chapter the following different storage schemes are used for matrices:
• conventional storage in a two-dimensional array;
• packed storage for symmetric, Hermitian or triangular matrices;
• rectangular full packed (RFP) storage for symmetric, Hermitian or triangular matrices;
• band storage for band matrices.
These storage schemes are compatible with those used in Chapter F06 (especially in the BLAS) and Chapter F08, but different schemes for packed or band storage are used in a few older routines in Chapters F01, F02, F03 and F04.
In the examples below, $*$ indicates an array element which need not be set and is not referenced by the routines. The examples illustrate only the relevant part of the arrays; array arguments may of course have additional rows or columns, according to the usual rules for passing array arguments in Fortran 77.

#### 3.3.1Conventional storage

The default scheme for storing matrices is the obvious one: a matrix $A$ is stored in a two-dimensional array a, with matrix element ${a}_{ij}$ stored in array element $\mathbf{a}\left(i,j\right)$.
If a matrix is triangular (upper or lower, as specified by the argument uplo), only the elements of the relevant triangle are stored; the remaining elements of the array need not be set. Such elements are indicated by * or $⌴$ in the examples below.
For example, when $n=4$:
uplo Triangular matrix $\mathbit{A}$ Storage in array a
'U' $\left(\begin{array}{llll}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ & {a}_{22}& {a}_{23}& {a}_{24}\\ & & {a}_{33}& {a}_{34}\\ & & & {a}_{44}\end{array}\right)$ $\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ \text{⌴}& {a}_{22}& {a}_{23}& {a}_{24}\\ \text{⌴}& \text{⌴}& {a}_{33}& {a}_{34}\\ \text{⌴}& \text{⌴}& \text{⌴}& {a}_{44}\end{array}$
'L' $\left(\begin{array}{llll}{a}_{11}& & & \\ {a}_{21}& {a}_{22}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& \\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right)$ $\begin{array}{cccc}{a}_{11}& \text{⌴}& \text{⌴}& \text{⌴}\\ {a}_{21}& {a}_{22}& \text{⌴}& \text{⌴}\\ {a}_{31}& {a}_{32}& {a}_{33}& \text{⌴}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}$
Routines which handle symmetric or Hermitian matrices allow for either the upper or lower triangle of the matrix (as specified by uplo) to be stored in the corresponding elements of the array; the remaining elements of the array need not be set.
For example, when $n=4$:
uplo Hermitian matrix $\mathbit{A}$ Storage in array a
'U' $\left(\begin{array}{llll}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ {\overline{a}}_{12}& {a}_{22}& {a}_{23}& {a}_{24}\\ {\overline{a}}_{13}& {\overline{a}}_{23}& {a}_{33}& {a}_{34}\\ {\overline{a}}_{14}& {\overline{a}}_{24}& {\overline{a}}_{34}& {a}_{44}\end{array}\right)$ $\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ \text{⌴}& {a}_{22}& {a}_{23}& {a}_{24}\\ \text{⌴}& \text{⌴}& {a}_{33}& {a}_{34}\\ \text{⌴}& \text{⌴}& \text{⌴}& {a}_{44}\end{array}$
'L' $\left(\begin{array}{llll}{a}_{11}& {\overline{a}}_{21}& {\overline{a}}_{31}& {\overline{a}}_{41}\\ {a}_{21}& {a}_{22}& {\overline{a}}_{32}& {\overline{a}}_{42}\\ {a}_{31}& {a}_{32}& {a}_{33}& {\overline{a}}_{43}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right)$ $\begin{array}{cccc}{a}_{11}& \text{⌴}& \text{⌴}& \text{⌴}\\ {a}_{21}& {a}_{22}& \text{⌴}& \text{⌴}\\ {a}_{31}& {a}_{32}& {a}_{33}& \text{⌴}\\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}$

#### 3.3.2Packed storage

Symmetric, Hermitian or triangular matrices may be stored more compactly, if the relevant triangle (again as specified by uplo) is packed by columns in a one-dimensional array. In this chapter, as in Chapters F06 and F08, arrays which hold matrices in packed storage, have names ending in P. For a matrix of order $n$, the array must have at least $n\left(n+1\right)/2$ elements. So:
• if $\mathbf{uplo}=\text{'U'}$, ${a}_{ij}$ is stored in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, ${a}_{ij}$ is stored in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $j\le i$.
For example:
uplo Triangle of matrix $\mathbit{A}$ Packed storage in array ap
'U' $\left(\begin{array}{llll}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ & {a}_{22}& {a}_{23}& {a}_{24}\\ & & {a}_{33}& {a}_{34}\\ & & & {a}_{44}\end{array}\right)$ ${a}_{11}\underbrace{{a}_{12}{a}_{22}}\underbrace{{a}_{13}{a}_{23}{a}_{33}}\underbrace{{a}_{14}{a}_{24}{a}_{34}{a}_{44}}$
'L' $\left(\begin{array}{llll}{a}_{11}& & & \\ {a}_{21}& {a}_{22}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& \\ {a}_{41}& {a}_{42}& {a}_{43}& {a}_{44}\end{array}\right)$ $\underbrace{{a}_{11}{a}_{21}{a}_{31}{a}_{41}}\underbrace{{a}_{22}{a}_{32}{a}_{42}}\underbrace{{a}_{33}{a}_{43}}{a}_{44}$
Note that for real symmetric matrices, packing the upper triangle by columns is equivalent to packing the lower triangle by rows; packing the lower triangle by columns is equivalent to packing the upper triangle by rows. (For complex Hermitian matrices, the only difference is that the off-diagonal elements are conjugated.)

#### 3.3.3Rectangular Full Packed (RFP) Storage

The rectangular full packed (RFP) storage format offers the same savings in storage as the packed storage format (described in Section 3.3.2), but is likely to be much more efficient in general since the block structure of the matrix is maintained. This structure can be exploited using block partition algorithms (see Section 2.8) in a similar way to matrices that use conventional storage.
Figure 1 gives a graphical representation of the key idea of RFP for the particular case of a lower triangular matrix of even dimensions. In all cases the original triangular matrix of stored elements is separated into a trapezoidal part and a triangular part. The number of columns in these two parts is equal when the dimension of the matrix is even, $n=2k$, while the trapezoidal part has $k+1$ columns when $n=2k+1$. The smaller part is then transposed and fitted onto the trapezoidal part forming a rectangle. The rectangle has dimensions $2k+1$ and $q$, where $q=k$ when $n$ is even and $q=k+1$ when $n$ is odd.
For routines using RFP there is the option of storing the rectangle as described above ($\mathbf{transr}=\text{'N'}$) or its transpose ($\mathbf{transr}=\text{'T'}$, for real a) or its conjugate transpose ($\mathbf{transr}=\text{'C'}$, for complex a).
As an example, we first consider RFP for the case $n=2k$ with $k=3$.
If $\mathbf{transr}=\text{'N'}$, then ar holds a as follows:
• For $\mathbf{uplo}=\text{'U'}$ the upper trapezoid $\mathbf{ar}\left(1:6,1:3\right)$ consists of the last three columns of a upper. The lower triangle $\mathbf{ar}\left(5:7,1:3\right)$ consists of the transpose of the first three columns of a upper.
• For $\mathbf{uplo}=\text{'L'}$ the lower trapezoid $\mathbf{ar}\left(2:7,1:3\right)$ consists of the first three columns of a lower. The upper triangle $\mathbf{ar}\left(1:3,1:3\right)$ consists of the transpose of the last three columns of a lower.
If $\mathbf{transr}=\text{'T'}$, then ar in both uplo cases is just the transpose of ar as defined when $\mathbf{transr}=\text{'N'}$.
uplo Triangle of matrix $\mathbit{A}$ Rectangular Full Packed matrix $\mathbit{AR}$
$\mathbf{transr}=\text{'N'}$ $\mathbf{transr}=\text{'T'}$
'U' $\left(\begin{array}{llllll}\mathbf{00}& \mathbf{01}& \mathbf{02}& 03& 04& 05\\ & \mathbf{11}& \mathbf{12}& 13& 14& 15\\ & & \mathbf{22}& 23& 24& 25\\ & & & 33& 34& 35\\ & & & & 44& 45\\ & & & & & 55\end{array}\right)$ $\begin{array}{ccc}03& 04& 05\\ 13& 14& 15\\ 23& 24& 25\\ 33& 34& 35\\ \mathbf{00}& 44& 45\\ \mathbf{01}& \mathbf{11}& 55\\ \mathbf{02}& \mathbf{12}& \mathbf{22}\end{array}$ $\begin{array}{ccccccc}03& 13& 23& 33& \mathbf{00}& \mathbf{01}& \mathbf{02}\\ 04& 14& 24& 34& 44& \mathbf{11}& \mathbf{12}\\ 05& 15& 25& 35& 45& 55& \mathbf{22}\end{array}$
'L' $\left(\begin{array}{l}00\\ 10& 11\\ 20& 21& 22\\ 30& 31& 32& \mathbf{33}\\ 40& 41& 42& \mathbf{43}& \mathbf{44}\\ 50& 51& 52& \mathbf{53}& \mathbf{54}& \mathbf{55}\end{array}\right)$ $\begin{array}{ccc}\mathbf{33}& \mathbf{43}& \mathbf{53}\\ 00& \mathbf{44}& \mathbf{54}\\ 10& 11& \mathbf{55}\\ 20& 21& 22\\ 30& 31& 32\\ 40& 41& 42\\ 50& 51& 52\end{array}$ $\begin{array}{ccccccc}\mathbf{33}& 00& 10& 20& 30& 40& 50\\ \mathbf{43}& \mathbf{44}& 11& 21& 31& 41& 51\\ \mathbf{53}& \mathbf{54}& \mathbf{55}& 22& 32& 42& 52\end{array}$
Now we consider RFP for the case $n=2k+1$ and $k=2$.
If $\mathbf{transr}=\text{'N'}$. ar holds a as follows:
• if $\mathbf{uplo}=\text{'U'}$ the upper trapezoid $\mathbf{ar}\left(1:5,1:3\right)$ consists of the last three columns of a upper. The lower triangle $\mathbf{ar}\left(4:5,1:2\right)$ consists of the transpose of the first two columns of a upper;
• if $\mathbf{uplo}=\text{'L'}$ the lower trapezoid $\mathbf{ar}\left(1:5,1:3\right)$ consists of the first three columns of a lower. The upper triangle $\mathbf{ar}\left(1:2,2:3\right)$ consists of the transpose of the last two columns of a lower.
If $\mathbf{transr}=\text{'T'}$. ar in both uplo cases is just the transpose of ar as defined when $\mathbf{transr}=\text{'N'}$.
uplo Triangle of matrix $\mathbit{A}$ Rectangular Full Packed matrix $\mathbit{AR}$
$\mathbf{transr}=\text{'N'}$ $\mathbf{transr}=\text{'T'}$
'U' $\left(\begin{array}{lllll}\mathbf{00}& \mathbf{01}& 02& 03& 04\\ & \mathbf{11}& 12& 13& 14\\ & & 22& 23& 24\\ & & & 33& 34\\ & & & & 44\end{array}\right)$ $\begin{array}{ccc}02& 03& 04\\ 12& 13& 14\\ 22& 23& 24\\ \mathbf{00}& 33& 34\\ \mathbf{01}& \mathbf{11}& 44\end{array}$ $\begin{array}{ccccc}02& 12& 22& \mathbf{00}& \mathbf{01}\\ 03& 13& 23& 33& \mathbf{11}\\ 04& 14& 24& 34& 44\end{array}$
'L' $\left(\begin{array}{l}00\\ 10& 11\\ 20& 21& 22\\ 30& 31& 32& \mathbf{33}\\ 40& 41& 42& \mathbf{43}& \mathbf{44}\end{array}\right)$ $\begin{array}{ccc}00& \mathbf{33}& \mathbf{43}\\ 10& 11& \mathbf{44}\\ 20& 21& 22\\ 30& 31& 32\\ 40& 41& 42\end{array}$ $\begin{array}{cccccc}00& 10& 20& 30& 40& 50\\ \mathbf{33}& 11& 21& 31& 41& 51\\ \mathbf{43}& \mathbf{44}& 22& 32& 42& 52\end{array}$
Explicitly, in the real matrix case, ar is a one-dimensional array of length $n\left(n+1\right)/2$ and contains the elements of a as follows:
for $\mathbf{uplo}=\text{'U'}$ and $\mathbf{transr}=\text{'N'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(\mathit{i}-1\right)+\mathit{j}+k+1\right)$, for $1\le \mathit{j}\le k$ and $1\le i\le \mathit{j}$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(\mathit{j}-k-1\right)+i\right)$, for $k and $1\le i\le j$;
for $\mathbf{uplo}=\text{'U'}$ and $\mathbf{transr}=\text{'T'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(j+k\right)+i\right)$, for $1\le j\le k$ and $1\le i\le j$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(i-1\right)+\mathit{j}-k\right)$, for $k and $1\le i\le j$;
for $\mathbf{uplo}=\text{'L'}$ and $\mathbf{transr}=\text{'N'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(j-1\right)+i+k-q+1\right)$, for $1\le j\le q$ and $j\le i\le n$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(i-k-1\right)+j-q\right)$, for $q and $j\le i\le n$;
for $\mathbf{uplo}=\text{'L'}$ and $\mathbf{transr}=\text{'T'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(i+k-q\right)+j\right)$, for $1\le j\le q$ and $1\le i\le n$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(j-1-q\right)+i-k\right)$, for $q and $1\le i\le n$.
In the case of complex matrices, the assumption is that the full matrix, if it existed, would be Hermitian. Thus, when $\mathbf{transr}=\text{'N'}$, the triangular portion of a that is, in the real case, transposed into the notional $\left(2k+1\right)×q$ RFP matrix is also conjugated. When $\mathbf{transr}=\text{'C'}$ the notional $q×\left(2k+1\right)$ RFP matrix is the conjugate transpose of the corresponding $\mathbf{transr}=\text{'N'}$ RFP matrix. Explicitly, for complex a, the array ar contains the elements (or conjugated elements) of a as follows:
for $\mathbf{uplo}=\text{'U'}$ and $\mathbf{transr}=\text{'N'}$,
${\overline{a}}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(\mathit{i}-1\right)+\mathit{j}+k+1\right)$, for $1\le \mathit{j}\le k$ and $1\le i\le \mathit{j}$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(\mathit{j}-k-1\right)+i\right)$, for $k and $1\le i\le j$;
for $\mathbf{uplo}=\text{'U'}$ and $\mathbf{transr}=\text{'C'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(j+k\right)+i\right)$, for $1\le j\le k$ and $1\le i\le j$, and
${\overline{a}}_{ij}$ is stored in $\mathbf{ar}\left(q\left(i-1\right)+\mathit{j}-k\right)$, for $k and $1\le i\le j$;
for $\mathbf{uplo}=\text{'L'}$ and $\mathbf{transr}=\text{'N'}$,
${a}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(j-1\right)+i+k-q+1\right)$, for $1\le j\le q$ and $j\le i\le n$, and
${\overline{a}}_{ij}$ is stored in $\mathbf{ar}\left(\left(2k+1\right)\left(i-k-1\right)+j-q\right)$, for $q and $j\le i\le n$;
for $\mathbf{uplo}=\text{'L'}$ and $\mathbf{transr}=\text{'C'}$,
${\overline{a}}_{ij}$ is stored in $\mathbf{ar}\left(q\left(i+k-q\right)+j\right)$, for $1\le j\le q$ and $1\le i\le n$, and
${a}_{ij}$ is stored in $\mathbf{ar}\left(q\left(j-1-q\right)+i-k\right)$, for $q and $1\le i\le n$.

#### 3.3.4Band storage

A band matrix with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals may be stored compactly in a two-dimensional array with ${k}_{l}+{k}_{u}+1$ rows and $n$ columns. Columns of the matrix are stored in corresponding columns of the array, and diagonals of the matrix are stored in rows of the array. This storage scheme should be used in practice only if ${k}_{l}$, ${k}_{u}\ll n$, although the routines in Chapters F07 and F08 work correctly for all values of ${k}_{l}$ and ${k}_{u}$. In Chapters F07 and F08 arrays which hold matrices in band storage have names ending in $\mathrm{B}$.
To be precise, elements of matrix elements ${a}_{ij}$ are stored as follows:
• ${a}_{ij}$ is stored in $\mathbf{ab}\left({k}_{u}+1+i-j,j\right)$ for $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{u}\right)\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{l}\right)$.
For example, when $n=5$, ${k}_{l}=2$ and ${k}_{u}=1$:
Band matrix $\mathbit{A}$ Band storage in array ab
$\left(\begin{array}{lllll}{a}_{11}& {a}_{12}& & & \\ {a}_{21}& {a}_{22}& {a}_{23}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}& \\ & {a}_{42}& {a}_{43}& {a}_{44}& {a}_{45}\\ & & {a}_{53}& {a}_{54}& {a}_{55}\end{array}\right)$ $\begin{array}{ccccc}& & & & \\ \text{*}& {a}_{12}& {a}_{23}& {a}_{34}& {a}_{45}\\ {a}_{11}& {a}_{22}& {a}_{33}& {a}_{44}& {a}_{55}\\ {a}_{21}& {a}_{32}& {a}_{43}& {a}_{54}& \text{*}\\ {a}_{31}& {a}_{42}& {a}_{53}& \text{*}& \text{*}\end{array}$
The elements marked $*$ in the upper left and lower right corners of the array ab need not be set, and are not referenced by the routines.
Note:  when a general band matrix is supplied for $LU$ factorization, space must be allowed to store an additional ${k}_{l}$ superdiagonals, generated by fill-in as a result of row interchanges. This means that the matrix is stored according to the above scheme, but with ${k}_{l}+{k}_{u}$ superdiagonals.
Triangular band matrices are stored in the same format, with either ${k}_{l}=0$ if upper triangular, or ${k}_{u}=0$ if lower triangular.
For symmetric or Hermitian band matrices with $k$ subdiagonals or superdiagonals, only the upper or lower triangle (as specified by uplo) need be stored:
• if $\mathbf{uplo}=\text{'U'}$, ${a}_{ij}$ is stored in $\mathbf{ab}\left(k+1+i-j,j\right)$ for $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, ${a}_{ij}$ is stored in $\mathbf{ab}\left(1+i-j,j\right)$ for $j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)$.
For example, when $n=5$ and $k=2$:
uplo Hermitian band matrix $\mathbit{A}$ Band storage in array ab
'U' $\left(\begin{array}{lllll}{a}_{11}& {a}_{12}& {a}_{13}& & \\ {\overline{a}}_{12}& {a}_{22}& {a}_{23}& {a}_{24}& \\ {\overline{a}}_{13}& {\overline{a}}_{23}& {a}_{33}& {a}_{34}& {a}_{35}\\ & {\overline{a}}_{24}& {\overline{a}}_{34}& {a}_{44}& {a}_{45}\\ & & {\overline{a}}_{35}& {\overline{a}}_{45}& {a}_{55}\end{array}\right)$ $\begin{array}{lllll}\text{*}& \text{*}& {a}_{13}& {a}_{24}& {a}_{35}\\ \text{*}& {a}_{12}& {a}_{23}& {a}_{34}& {a}_{45}\\ {a}_{11}& {a}_{22}& {a}_{33}& {a}_{44}& {a}_{55}\end{array}$
'L' $\left(\begin{array}{lllll}{a}_{11}& {\overline{a}}_{21}& {\overline{a}}_{31}& & \\ {a}_{21}& {a}_{22}& {\overline{a}}_{32}& {\overline{a}}_{42}& \\ {a}_{31}& {a}_{32}& {a}_{33}& {\overline{a}}_{43}& {\overline{a}}_{53}\\ & {a}_{42}& {a}_{43}& {a}_{44}& {\overline{a}}_{54}\\ & & {a}_{53}& {a}_{54}& {a}_{55}\end{array}\right)$ $\begin{array}{lllll}{a}_{11}& {a}_{22}& {a}_{33}& {a}_{44}& {a}_{55}\\ {a}_{21}& {a}_{32}& {a}_{43}& {a}_{54}& \text{*}\\ {a}_{31}& {a}_{42}& {a}_{53}& \text{*}& \text{*}\end{array}$
Note that different storage schemes for band matrices are used by some routines in Chapters F01, F02, F03 and F04.

#### 3.3.5Unit triangular matrices

Some routines in this chapter have an option to handle unit triangular matrices (that is, triangular matrices with diagonal elements $\text{}=1$). This option is specified by an argument diag. If $\mathbf{diag}=\text{'U'}$ (Unit triangular), the diagonal elements of the matrix need not be stored, and the corresponding array elements are not referenced by the routines. The storage scheme for the rest of the matrix (whether conventional, packed or band) remains unchanged.

#### 3.3.6Real diagonal elements of complex matrices

Complex Hermitian matrices have diagonal elements that are by definition purely real. In addition, complex triangular matrices which arise in Cholesky factorization are defined by the algorithm to have real diagonal elements.
If such matrices are supplied as input to routines in Chapters F07 and F08, the imaginary parts of the diagonal elements are not referenced, but are assumed to be zero. If such matrices are returned as output by the routines, the computed imaginary parts are explicitly set to zero.

### 3.4Parameter Conventions

#### 3.4.1Option arguments

Most routines in this chapter have one or more option arguments, of type CHARACTER. The descriptions in Section 5 of the routine documents refer only to upper-case values (for example $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$); however, in every case, the corresponding lower-case characters may be supplied (with the same meaning). Any other value is illegal.
A longer character string can be passed as the actual argument, making the calling program more readable, but only the first character is significant. (This is a feature of Fortran 77.) For example:
`Call dgetrs('Transpose',...)`

#### 3.4.2Problem dimensions

It is permissible for the problem dimensions (for example, m in f07adf, n or nrhs in f07aef) to be passed as zero, in which case the computation (or part of it) is skipped. Negative dimensions are regarded as an error.

#### 3.4.3Length of work arrays

A few routines implementing block partitioned algorithms require workspace sufficient to hold one block of rows or columns of the matrix if they are to achieve optimum levels of performance — for example, workspace of size $n×nb$, where $nb$ is the optimum block size. In such cases, the actual declared length of the work array must be passed as a separate argument lwork, which immediately follows work in the argument-list.
The routine will still perform correctly when less workspace is provided: it uses the largest block size allowed by the amount of workspace supplied, as long as this is likely to give better performance than the unblocked algorithm. On exit, $\mathbf{work}\left(1\right)$ contains the minimum value of lwork which would allow the routine to use the optimum block size; this value of lwork may be used for subsequent runs.
If lwork indicates that there is insufficient workspace to perform the unblocked algorithm, this is regarded as an illegal value of lwork, and is treated like any other illegal argument value (see Section 3.4.4), though $\mathbf{work}\left(1\right)$ will still be set as described above.
If you are in doubt how much workspace to supply and are concerned to achieve optimum performance, supply a generous amount (assume a block size of $64$, say), and then examine the value of $\mathbf{work}\left(1\right)$ on exit.

#### 3.4.4Error-handling and the diagnostic argument INFO

Routines in this chapter do not use the usual NAG Library error-handling mechanism, involving the argument IFAIL. Instead they have a diagnostic argument info. (Thus they preserve complete compatibility with the LAPACK specification.)
Whereas IFAIL is an Input/Output argument and must be set before calling a routine, info is purely an Output argument and need not be set before entry.
info indicates the success or failure of the computation, as follows:
• $\mathbf{info}=0$: successful termination
• $\mathbf{info}>0$: failure in the course of computation, control returned to the calling program
If the routine document specifies that the routine may terminate with $\mathbf{info}>0$, then it is essential to test info on exit from the routine. (This corresponds to a soft failure in terms of the usual NAG error-handling terminology.) No error message is output.
All routines check that input arguments such as n or lda or option arguments of type CHARACTER have permitted values. If an illegal value of the $i$th argument is detected, info is set to $-i$, a message is output, and execution of the program is terminated. (This corresponds to a hard failure in the usual NAG terminology.)

### 3.5Tables of Driver and Computational Routines

#### 3.5.1Real matrices

Each entry in the following tables, listing real matrices, gives:
• the NAG routine name and
• the double precision LAPACK routine name.
Table 1
Routines for real general matrices
Type of matrix and storage scheme
Operation general general band general tridiagonal
driver f07aaf f07baf f07caf
expert driver f07abf f07bbf f07cbf
mixed precision driver f07acf
solve f07aef f07bef f07cef
scaling factors f07aff f07bff
condition number f07agf f07bgf f07cgf
error estimate f07ahf f07bhf f07chf
invert f07ajf
Table 2
Routines for real symmetric positive definite and positive semidefinite matrices
Type of matrix and storage scheme
Operation symmetric positive definite symmetric positive definite (packed storage) symmetric positive definite (RFP storage) symmetric positive definite band symmetric positive definite tridiagonal symmetric positive semidefinite
driver f07faf f07gaf f07haf f07jaf
expert driver f07fbf f07gbf f07hbf f07jbf
mixed precision f07fcf
factorize f07fdf f07gdf f07wdf f07hdf f07jdf f07kdf
solve f07fef f07gef f07wef f07hef f07jef
scaling factors f07fff f07gff f07hff
condition number f07fgf f07ggf f07hgf f07jgf
error estimate f07fhf f07ghf f07hhf f07jhf
invert f07fjf f07gjf f07wjf
Table 3
Routines for real symmetric indefinite matrices
Type of matrix and storage scheme
Operation symmetric indefinite symmetric indefinite (packed storage)
driver f07maf f07paf
expert driver f07mbf f07pbf
factorize f07mdf f07pdf
solve f07mef f07pef
condition number f07mgf f07pgf
error estimate f07mhf f07phf
invert f07mjf f07pjf
Table 4
Routines for real triangular matrices
Type of matrix and storage scheme
Operation triangular triangular (packed storage) triangular (RFP storage) triangular band
solve f07tef f07uef f07vef
condition number f07tgf f07ugf f07vgf
error estimate f07thf f07uhf f07vhf
invert f07tjf f07ujf f07wkf

#### 3.5.2Complex matrices

Each entry in the following tables, listing complex matrices, gives:
• the NAG routine name and
• the double precision LAPACK routine name.
Table 5
Routines for complex general matrices
Type of matrix and storage scheme
Operation general general band general tridiagonal
driver f07anf f07bnf f07cnf
expert driver f07apf f07bpf f07cpf
mixed precision driver f07aqf
factorize f07arf f07brf f07crf
solve f07asf f07bsf f07csf
scaling factors f07atf f07btf
condition number f07auf f07buf f07cuf
error estimate f07avf f07bvf f07cvf
invert f07awf
Table 6
Routines for complex Hermitian positive definite and positive semidefinite matrices
Type of matrix and storage scheme
Operation Hermitian positive definite Hermitian positive definite (packed storage) Hermitian positive definite (RFP storage) Hermitian positive definite band Hermitian positive definite tridiagonal Hermitian positive semidefinite
driver f07fnf f07gnf f07hnf f07jnf
expert driver f07fpf f07gpf f07hpf f07jpf
mixed precision driver f07fqf
factorize f07frf f07grf f07wrf f07hrf f07jrf f07krf
solve f07fsf f07gsf f07wsf f07hsf f07jsf
scaling factors f07ftf f07gtf
condition number f07fuf f07guf f07huf f07juf
error estimate f07fvf f07gvf f07hvf f07jvf
invert f07fwf f07gwf f07wwf
Table 7
Routines for complex Hermitian and symmetric indefinite matrices
Type of matrix and storage scheme
Operation Hermitian indefinite symmetric indefinite (packed storage) Hermitian indefinite band symmetric indefinite tridiagonal
driver f07mnf f07nnf f07pnf f07qnf
expert driver f07mpf f07npf f07ppf f07qpf
factorize f07mrf f07nrf f07prf f07qrf
solve f07msf f07nsf f07psf f07qsf
condition number f07muf f07nuf f07puf f07quf
error estimate f07mvf f07nvf f07pvf f07qvf
invert f07mwf f07nwf f07pwf f07qwf
Table 8
Routines for complex triangular matrices
Type of matrix and storage scheme
Operation triangular triangular (packed storage) triangular (RFP storage) triangular band
solve f07tsf f07usf f07vsf
condition number f07tuf f07uuf f07vuf
error estimate f07tvf f07uvf f07vvf
invert f07twf f07uwf f07wxf

## 4Functionality Index

 Apply iterative refinement to the solution and compute error estimates,
 after factorizing the matrix of coefficients,
 complex band matrix f07bvf
 complex Hermitian indefinite matrix f07mvf
 complex Hermitian indefinite matrix, packed storage f07pvf
 complex Hermitian positive definite band matrix f07hvf
 complex Hermitian positive definite matrix f07fvf
 complex Hermitian positive definite matrix, packed storage f07gvf
 complex Hermitian positive definite tridiagonal matrix f07jvf
 complex matrix f07avf
 complex symmetric indefinite matrix f07nvf
 complex symmetric indefinite matrix, packed storage f07qvf
 complex tridiagonal matrix f07cvf
 real band matrix f07bhf
 real matrix f07ahf
 real symmetric indefinite matrix f07mhf
 real symmetric indefinite matrix, packed storage f07phf
 real symmetric positive definite band matrix f07hhf
 real symmetric positive definite matrix f07fhf
 real symmetric positive definite matrix, packed storage f07ghf
 real symmetric positive definite tridiagonal matrix f07jhf
 real tridiagonal matrix f07chf
 Compute error estimates,
 complex triangular band matrix f07vvf
 complex triangular matrix f07tvf
 complex triangular matrix, packed storage f07uvf
 real triangular band matrix f07vhf
 real triangular matrix f07thf
 real triangular matrix, packed storage f07uhf
 Compute row and column scalings,
 complex band matrix f07btf
 complex Hermitian positive definite band matrix f07htf
 complex Hermitian positive definite matrix f07ftf
 complex Hermitian positive definite matrix, packed storage f07gtf
 complex matrix f07atf
 real band matrix f07bff
 real matrix f07aff
 real symmetric positive definite band matrix f07hff
 real symmetric positive definite matrix f07fff
 real symmetric positive definite matrix, packed storage f07gff
 Condition number estimation,
 after factorizing the matrix of coefficients,
 complex band matrix f07buf
 complex Hermitian indefinite matrix f07muf
 complex Hermitian indefinite matrix, packed storage f07puf
 complex Hermitian positive definite band matrix f07huf
 complex Hermitian positive definite matrix f07fuf
 complex Hermitian positive definite matrix, packed storage f07guf
 complex Hermitian positive definite tridiagonal matrix f07juf
 complex matrix f07auf
 complex symmetric indefinite matrix f07nuf
 complex symmetric indefinite matrix, packed storage f07quf
 complex tridiagonal matrix f07cuf
 real band matrix f07bgf
 real matrix f07agf
 real symmetric indefinite matrix f07mgf
 real symmetric indefinite matrix, packed storage f07pgf
 real symmetric positive definite band matrix f07hgf
 real symmetric positive definite matrix f07fgf
 real symmetric positive definite matrix, packed storage f07ggf
 real symmetric positive definite tridiagonal matrix f07jgf
 real tridiagonal matrix f07cgf
 complex triangular band matrix f07vuf
 complex triangular matrix f07tuf
 complex triangular matrix, packed storage f07uuf
 real triangular band matrix f07vgf
 real triangular matrix f07tgf
 real triangular matrix, packed storage f07ugf
 $LD{L}^{\mathrm{T}}$ factorization,
 complex Hermitian positive definite tridiagonal matrix f07jrf
 real symmetric positive definite tridiagonal matrix f07jdf
 $L{L}^{\mathrm{T}}$ or ${U}^{\mathrm{T}}U$ factorization,
 complex Hermitian positive definite band matrix f07hrf
 complex Hermitian positive definite matrix f07frf
 complex Hermitian positive definite matrix, packed storage f07grf
 complex Hermitian positive definite matrix, RFP storage f07wrf
 complex Hermitian positive semidefinite matrix f07krf
 real symmetric positive definite band matrix f07hdf
 real symmetric positive definite matrix f07fdf
 real symmetric positive definite matrix, packed storage f07gdf
 real symmetric positive definite matrix, RFP storage f07wdf
 real symmetric positive semidefinite matrix f07kdf
 $LU$ factorization,
 complex band matrix f07brf
 complex matrix f07arf
 complex tridiagonal matrix f07crf
 real band matrix f07bdf
 real tridiagonal matrix f07cdf
 Matrix inversion,
 after factorizing the matrix of coefficients,
 complex Hermitian indefinite matrix f07mwf
 complex Hermitian indefinite matrix, packed storage f07pwf
 complex Hermitian positive definite matrix f07fwf
 complex Hermitian positive definite matrix, packed storage f07gwf
 complex Hermitian positive definite matrix, RFP storage f07wwf
 complex matrix f07awf
 complex symmetric indefinite matrix f07nwf
 complex symmetric indefinite matrix, packed storage f07qwf
 real matrix f07ajf
 real symmetric indefinite matrix f07mjf
 real symmetric indefinite matrix, packed storage f07pjf
 real symmetric positive definite matrix f07fjf
 real symmetric positive definite matrix, packed storage f07gjf
 real symmetric positive definite matrix, RFP storage f07wjf
 complex triangular matrix f07twf
 complex triangular matrix, packed storage f07uwf
 complex triangular matrix, RFP storage,
 expert driver f07wxf
 real triangular matrix f07tjf
 real triangular matrix, packed storage f07ujf
 real triangular matrix, RFP storage,
 expert driver f07wkf
 $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ or $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ factorization,
 complex Hermitian indefinite matrix f07mrf
 complex Hermitian indefinite matrix, packed storage f07prf
 complex symmetric indefinite matrix f07nrf
 complex symmetric indefinite matrix, packed storage f07qrf
 real symmetric indefinite matrix f07mdf
 real symmetric indefinite matrix, packed storage f07pdf
 Solution of simultaneous linear equations,
 after factorizing the matrix of coefficients,
 complex band matrix f07bsf
 complex Hermitian indefinite matrix f07msf
 complex Hermitian indefinite matrix, packed storage f07psf
 complex Hermitian positive definite band matrix f07hsf
 complex Hermitian positive definite matrix f07fsf
 complex Hermitian positive definite matrix, packed storage f07gsf
 complex Hermitian positive definite matrix, RFP storage f07wsf
 complex Hermitian positive definite tridiagonal  matrix f07jsf
 complex matrix f07asf
 complex symmetric indefinite matrix f07nsf
 complex symmetric indefinite matrix, packed storage f07qsf
 complex tridiagonal matrix f07csf
 real band matrix f07bef
 real matrix f07aef
 real symmetric indefinite matrix f07mef
 real symmetric indefinite matrix, packed storage f07pef
 real symmetric positive definite band matrix f07hef
 real symmetric positive definite matrix f07fef
 real symmetric positive definite matrix, packed storage f07gef
 real symmetric positive definite matrix, RFP storage f07wef
 real symmetric positive definite tridiagonal matrix f07jef
 real tridiagonal matrix f07cef
 expert drivers (with condition and error estimation):
 complex band matrix f07bpf
 complex Hermitian indefinite matrix f07mpf
 complex Hermitian indefinite matrix, packed storage f07ppf
 complex Hermitian positive definite band matrix f07hpf
 complex Hermitian positive definite matrix f07fpf
 complex Hermitian positive definite matrix, packed storage f07gpf
 complex Hermitian positive definite tridiagonal matrix f07jpf
 complex matrix f07apf
 complex symmetric indefinite matrix f07npf
 complex symmetric indefinite matrix, packed storage f07qpf
 complex tridiagonal matrix f07cpf
 real band matrix f07bbf
 real matrix f07abf
 real symmetric indefinite matrix f07mbf
 real symmetric indefinite matrix, packed storage f07pbf
 real symmetric positive definite band matrix f07hbf
 real symmetric positive definite matrix f07fbf
 real symmetric positive definite matrix, packed storage f07gbf
 real symmetric positive definite tridiagonal matrix f07jbf
 real tridiagonal matrix f07cbf
 simple drivers,
 complex band matrix f07bnf
 complex Hermitian indefinite matrix f07mnf
 complex Hermitian indefinite matrix, packed storage f07pnf
 complex Hermitian positive definite band matrix f07hnf
 complex Hermitian positive definite matrix f07fnf
 complex Hermitian positive definite matrix, packed storage f07gnf
 complex Hermitian positive definite matrix, using mixed precision f07fqf
 complex Hermitian positive definite tridiagonal matrix f07jnf
 complex matrix f07anf
 complex matrix, using mixed precision f07aqf
 complex symmetric indefinite matrix f07nnf
 complex symmetric indefinite matrix, packed storage f07qnf
 complex triangular band matrix f07vsf
 complex triangular matrix f07tsf
 complex triangular matrix, packed storage f07usf
 complex tridiagonal matrix f07cnf
 real band matrix f07baf
 real matrix f07aaf
 real matrix, using mixed precision f07acf
 real symmetric indefinite matrix f07maf
 real symmetric indefinite matrix, packed storage f07paf
 real symmetric positive definite band matrix f07haf
 real symmetric positive definite matrix f07faf
 real symmetric positive definite matrix, packed storage f07gaf
 real symmetric positive definite matrix, using mixed precision f07fcf
 real symmetric positive definite tridiagonal matrix f07jaf
 real triangular band matrix f07vef
 real triangular matrix f07tef
 real triangular matrix, packed storage f07uef
 real tridiagonal matrix f07caf

None.

None.

## 7References

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1988) Algorithm 674: Fortran codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford