F07 Chapter Contents
F07 Chapter Introduction (PDF version)
NAG Library Manual

NAG Library Chapter Introduction

F07 – Linear Equations (LAPACK)

+ Contents

1  Scope of the Chapter

This chapter provides routines for the solution of systems of simultaneous linear equations, and associated computations. It provides routines for 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 specialized 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.

2  Background 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.

2.1  Notation

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:
x^ a computed solution to Ax=b, (which usually differs from the exact solution x because of round-off error)
r = b-Ax^   the residual corresponding to the computed solution x^
x= maxi xi   the -norm of the vector x
x1 = j=1 n xj   the 1-norm of the vector x
A = maxi j aij   the -norm of the matrix A
A1 = maxj i=1 n aij   the 1-norm of the matrix A
x the vector with elements xi
A the matrix with elements aij
Inequalities of the form AB are interpreted component-wise, that is aijbij for all i,j.

2.2  Matrix 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 by 2 unit matrices on the diagonal corresponding to the 2 by 2 blocks of D. The permutation matrix P (which represents symmetric row-and-column interchanges) and the 2 by 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=UDUT or A=LDLT with diagonal D, which is simply a variant form of the Cholesky factorization.

2.3  Solution 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

2.4  Sensitivity and Error Analysis

2.4.1  Normwise 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+δx is the exact solution to a perturbed problem A+δAx+δx=b+δb, then
δx x κA δA A +δb b +second-order terms
where κA is the condition number of A defined by
κA = A.A-1 . (3)
In other words, relative errors in A or b may be amplified in x by a factor κA. Section 2.4.2 discusses how to compute or estimate κA.
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 δA A  and δb b  are usually at most pnε, where ε is the machine precision and pn is an increasing function of n which is seldom larger than 10n (although in theory it can be as large as 2n-1).
In other words, the computed solution x^ is the exact solution of a linear system A+δAx^=b+δb which is close to the original system in a normwise sense.

2.4.2  Estimating condition numbers

The previous section has emphasised the usefulness of the quantity κA in understanding the sensitivity of the solution of Ax=b. To compute the value of κA from equation (3) is more expensive than solving Ax=b in the first place. Hence it is standard practice to estimate κA, in either the 1-norm or the -norm, by a method which only requires On2 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 κA which is important rather than its precise value.
Because κA is infinite if A is singular, the routines in this chapter actually return the reciprocal of κA.

2.4.3  Scaling and Equilibration

The condition of a matrix and hence the accuracy of the computed solution, may be improved by scaling; thus if D1 and D2 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.

2.4.4  Componentwise 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 ω is given by
ω= maxi ri A. x^+bi .
Routines are provided in this chapter which compute ω, 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 is estimated cheaply (without computing A-1) by a modification of the method used to estimate κA.

2.4.5  Iterative refinement of the solution

If x^ is an approximate computed solution to Ax=b, and r is the corresponding residual, then a procedure for iterative refinement of x^ can be defined as follows, starting with x0=x^:
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.

2.5  Matrix 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-1b; 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.

2.6  Packed 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 nn+1/2 or a two-dimensional array with nn+1/2 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.

2.7  Band 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 kl subdiagonals and ku superdiagonals, then L is not a band matrix but still has at most kl nonzero elements below the diagonal in each column; and U has at most kl+ku 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.

2.8  Block 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 parameter, 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.

2.9  Mixed 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.

3  Recommendations on Choice and Use of Available Routines

3.1  Available 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:
For each of the above types of matrix (except where indicated), routines are provided to perform the following computations:
(a) (except for RFP matrices) solve a system of linear equations (driver routines);
(b) (except for RFP matrices) solve a system of linear equations with condition and error estimation (expert drivers);
(c) (except for triangular matrices) factorize the matrix (see Section 2.2);
(d) solve a system of linear equations, using the factorization (see Section 2.3);
(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;
(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);
(g) (except for band and tridiagonal matrices) invert the matrix, using the factorization (see Section 2.5);
(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.

3.2  NAG 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 (DGETRF).
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.

3.3  Matrix Storage Schemes

In this chapter the following different storage schemes are used for 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.

3.3.1  Conventional 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 aij stored in array element Ai,j .
If a matrix is triangular (upper or lower, as specified by the parameter 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 * in the examples below.
For example, when n=4:
UPLO Triangular matrix A Storage in array A
'U' a11 a12 a13 a14 a22 a23 a24 a33 a34 a44   a11 a12 a13 a14 * a22 a23 a24 * * a33 a34 * * * a44  
'L' a11 a21 a22 a31 a32 a33 a41 a42 a43 a44   a11 * * * a21 a22 * * a31 a32 a33 * a41 a42 a43 a44  
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 A Storage in array A
'U' a11 a12 a13 a14 a-12 a22 a23 a24 a-13 a-23 a33 a34 a-14 a-24 a-34 a44   a11 a12 a13 a14 * a22 a23 a24 * * a33 a34 * * * a44  
'L' a11 a-21 a-31 a-41 a21 a22 a-32 a-42 a31 a32 a33 a-43 a41 a42 a43 a44   a11 * * * a21 a22 * * a31 a32 a33 * a41 a42 a43 a44  

3.3.2  Packed 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 nn+1/2 elements. So:
For example:
  Triangle of matrix A Packed storage in array AP
UPLO='U' a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 a11 a12 a22 a13 a23 a33 a14 a24 a34 a44
UPLO='L' a11 a21 a22 a31 a32 a33 a41 a42 a43 a44 a11 a21 a31 a41 a22 a32 a42 a33 a43 a44
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.)

3.3.3  Rectangular 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 f07intro_rfp
Figure 1
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 (TRANSR='N') or its transpose (TRANSR='T').
As an example, we first consider RFP for the case n=2k with k=3.
If TRANSR='N', then AF holds A as follows:
If TRANSR='T', then AF in both UPLO cases is just the transpose of AF as defined when TRANSR='N'.
UPLO Triangle of matrix A Rectangular Full Packed matrix AF
TRANSR='N' TRANSR='T'
'U' 00 01 02 03 04 05 11 12 13 14 15 22 23 24 25 33 34 35 44 45 55   03 04 05 13 14 15 23 24 25 33 34 35 00 44 45 01 11 55 02 12 22   03 13 23 33 00 01 02 04 1424 3444 11 12 05 1525 35 4555 22  
'L' 00 10 11 20 21 22 30 31 32 33 40 41 42 43 44 50 51 52 53 54 55   33 43 53 00 44 54 10 11 55 20 21 22 30 31 32 40 41 42 50 51 52   33 00 10 20 30 40 50 43 44 11 21 31 41 51 53 54 55 22 32 42 52  
Now we consider RFP for the case n=2k+1 and k=2.
If TRANSR='N'. AF holds A as follows:
If TRANSR='T'. AF in both UPLO cases is just the transpose of AF as defined when TRANSR='N'.
UPLO Triangle of matrix A Rectangular Full Packed matrix AF
TRANSR='N' TRANSR='T'
'U' 00 01 02 03 04 11 12 13 14 22 23 24 33 34 44   02 03 04 12 13 14 22 23 24 00 33 34 01 11 44   02 12 22 00 01 03 13 23 33 11 04 14 24 34 44  
'L' 00 10 11 20 21 22 30 31 32 33 40 41 42 43 44   00 33 43 10 11 44 20 21 22 30 31 32 40 41 42   00 10 20 30 40 50 33 11 21 31 41 51 43 44 22 32 42 52  
Explicitly, AF is a one-dimensional array of length nn+1/2 and contains the elements of A as follows:

3.3.4  Band storage

A band matrix with kl subdiagonals and ku superdiagonals may be stored compactly in a two-dimensional array with kl+ku+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 kl, kun, although the routines in Chapters F07 and F08 work correctly for all values of kl and ku. In Chapters F07 and F08 arrays which hold matrices in band storage have names ending in B.
To be precise, elements of matrix elements aij are stored as follows:
For example, when n=5, kl=2 and ku=1:
Band matrix A Band storage in array AB
a11 a12 a21 a22 a23 a31 a32 a33 a34 a42 a43 a44 a45 a53 a54 a55 * a12 a23 a34 a45 a11 a22 a33 a44 a55 a21 a32 a43 a54 * a31 a42 a53 * *
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 kl 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 kl+ku superdiagonals.
Triangular band matrices are stored in the same format, with either kl=0 if upper triangular, or ku=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:
For example, when n=5 and k=2:
UPLO Hermitian band matrix A Band storage in array AB
'U' a11 a12 a13 a-12 a22 a23 a24 a-13 a-23 a33 a34 a35 a-24 a-34 a44 a45 a-35 a-45 a55 * * a13 a24 a35 * a12 a23 a34 a45 a11 a22 a33 a44 a55
'L' a11 a-21 a-31 a21 a22 a-32 a-42 a31 a32 a33 a-43 a-53 a42 a43 a44 a-54 a53 a54 a55 a11 a22 a33 a44 a55 a21 a32 a43 a54 * a31 a42 a53 * *
Note that different storage schemes for band matrices are used by some routines in Chapters F01, F02, F03 and F04.

3.3.5  Unit triangular matrices

Some routines in this chapter have an option to handle unit triangular matrices (that is, triangular matrices with diagonal elements =1). This option is specified by an argument DIAG. If DIAG='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.

3.3.6  Real 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.

3.4  Parameter Conventions

3.4.1  Option parameters

Most routines in this chapter have one or more option parameters, of type CHARACTER. The descriptions in Section 5 of the routine documents refer only to upper-case values (for example UPLO='U' or '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 parameter, 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',...)

3.4.2  Problem dimensions

It is permissible for the problem dimensions (for example, M in F07ADF (DGETRF), N or NRHS in F07AEF (DGETRS)) to be passed as zero, in which case the computation (or part of it) is skipped. Negative dimensions are regarded as an error.

3.4.3  Length 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 parameter LWORK, which immediately follows WORK in the parameter-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, WORK1 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 parameter value (see Section 3.4.4), though WORK1 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 WORK1 on exit.

3.4.4  Error-handling and the diagnostic parameter INFO

Routines in this chapter do not use the usual NAG Library error-handling mechanism, involving the parameter IFAIL. Instead they have a diagnostic parameter INFO. (Thus they preserve complete compatibility with the LAPACK specification.)
Whereas IFAIL is an Input/Output parameter and must be set before calling a routine, INFO is purely an Output parameter and need not be set before entry.
INFO indicates the success or failure of the computation, as follows:
If the routine document specifies that the routine may terminate with 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 parameters such as N or LDA or option parameters of type CHARACTER have permitted values. If an illegal value of the ith parameter 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.)

3.5  Tables of Available Computational Routines

3.5.1  Real matrices

Each entry in the following tables, listing real matrices, gives:
Type of matrix and storage scheme general general band general tridiagonal
factorize F07ADF (DGETRF) F07BDF (DGBTRF) F07CDF (DGTTRF)
solve F07AEF (DGETRS) F07BEF (DGBTRS) F07CEF (DGTTRS)
scaling factors F07AFF (DGEEQU) F07BFF (DGBEQU)  
condition number F07AGF (DGECON) F07BGF (DGBCON) F07CGF (DGTCON)
error estimate F07AHF (DGERFS) F07BHF (DGBRFS) F07CHF (DGTRFS)
invert F07AJF (DGETRI)    
Table 1
Routines for real general matrices
Type of matrix and storage scheme symmetric positive definite symmetric positive definite (packed storage) symmetric positive definite (RFP storage) symmetric positive definite band symmetric positive definite tridiagonal symmetric positive semidefinite factorize
factorize F07FDF (DPOTRF) F07GDF (DPPTRF) F07WDF (DPFTRF) F07HDF (DPBTRF) F07JDF (DPTTRF)  
solve F07FEF (DPOTRS) F07GEF (DPPTRS) F07WEF (DPFTRS) F07HEF (DPBTRS) F07JEF (DPTTRS)  
scaling factors F07FFF (DPOEQU) F07GFF (DPPEQU)   F07HFF (DPBEQU)    
condition number F07FGF (DPOCON) F07GGF (DPPCON)   F07HGF (DPBCON) F07JGF (DPTCON)  
error estimate F07FHF (DPORFS) F07GHF (DPPRFS)   F07HHF (DPBRFS) F07JHF (DPTRFS)  
invert F07FJF (DPOTRI) F07GJF (DPPTRI) F07WJF (DPFTRI)      
semidefinite           F07KDF (DPSTRF)
Table 2
Routines for real symmetric positive definite matrices
Type of matrix and storage scheme symmetric indefinite symmetric indefinite (packed storage)
factorize F07MDF (DSYTRF) F07PDF (DSPTRF)
solve F07MEF (DSYTRS) F07PEF (DSPTRS)
condition number F07MGF (DSYCON) F07PGF (DSPCON)
error estimate F07MHF (DSYRFS) F07PHF (DSPRFS)
invert F07MJF (DSYTRI) F07PJF (DSPTRI)
Table 3
Routines for real symmetric indefinite matrices
Type of matrix and storage scheme triangular triangular (packed storage) triangular (RFP storage) triangular band
solve F07TEF (DTRTRS) F07UEF (DTPTRS)   F07VEF (DTBTRS)
condition number F07TGF (DTRCON) F07UGF (DTPCON)   F07VGF (DTBCON)
error estimate F07THF (DTRRFS) F07UHF (DTPRFS)   F07VHF (DTBRFS)
invert F07TJF (DTRTRI) F07UJF (DTPTRI) F07WKF (DTFTRI)  
Table 4
Routines for real triangular matrices

3.5.2  Complex matrices

Each entry in the following tables, listing complex matrices, gives:
Type of matrix and storage scheme general general band general tridiagonal
factorize F07ARF (ZGETRF) F07BRF (ZGBTRF) F07CRF (ZGTTRF)
solve F07ASF (ZGETRS) F07BSF (ZGBTRS) F07CSF (ZGTTRS)
scaling factors F07ATF (ZGEEQU) F07BTF (ZGBEQU)  
condition number F07AUF (ZGECON) F07BUF (ZGBCON) F07CUF (ZGTCON)
error estimate F07AVF (ZGERFS) F07BVF (ZGBRFS) F07CVF (ZGTRFS)
invert F07AWF (ZGETRI)    
Table 5
Routines for complex general matrices
Type of matrix and storage scheme Hermitian positive definite Hermitian positive definite (packed storage) Hermitian positive definite (RFP storage) Hermitian positive definite band Hermitian positive definite tridiagonal Hermitian positive semidefinite factorize
factorize F07FRF (ZPOTRF) F07GRF (ZPPTRF) F07WRF (ZPFTRF) F07HRF (ZPBTRF) F07JRF (ZPTTRF)  
solve F07FSF (ZPOTRS) F07GSF (ZPPTRS) F07WSF (ZPFTRS) F07HSF (ZPBTRS) F07JSF (ZPTTRS)  
scaling factors F07FTF (ZPOEQU) F07GTF (ZPPEQU)        
condition number F07FUF (ZPOCON) F07GUF (ZPPCON)   F07HUF (ZPBCON) F07JUF (ZPTCON)  
error estimate F07FVF (ZPORFS) F07GVF (ZPPRFS)   F07HVF (ZPBRFS) F07JVF (ZPTRFS)  
invert F07FWF (ZPOTRI) F07GWF (ZPPTRI) F07WWF (ZPFTRI)      
semidefinite matrices           F07KRF (ZPSTRF)
Table 6
Routines for complex Hermitian positive definite matrices
Type of matrix and storage scheme Hermitian indefinite symmetric indefinite (packed storage) Hermitian indefinite band symmetric indefinite tridiagonal
factorize F07MRF (ZHETRF) F07NRF (ZSYTRF) F07PRF (ZHPTRF) F07QRF (ZSPTRF)
solve F07MSF (ZHETRS) F07NSF (ZSYTRS) F07PSF (ZHPTRS) F07QSF (ZSPTRS)
condition number F07MUF (ZHECON) F07NUF (ZSYCON) F07PUF (ZHPCON) F07QUF (ZSPCON)
error estimate F07MVF (ZHERFS) F07NVF (ZSYRFS) F07PVF (ZHPRFS) F07QVF (ZSPRFS)
invert F07MWF (ZHETRI) F07NWF (ZSYTRI) F07PWF (ZHPTRI) F07QWF (ZSPTRI)
Table 7
Routines for complex Hermitian and symmetric indefinite matrices
Type of matrix and storage scheme triangular triangular (packed storage) triangular (RFP storage) triangular band
solve F07TSF (ZTRTRS) F07USF (ZTPTRS)   F07VSF (ZTBTRS)
condition number F07TUF (ZTRCON) F07UUF (ZTPCON)   F07VUF (ZTBCON)
error estimate F07TVF (ZTRRFS) F07UVF (ZTPRFS)   F07VVF (ZTBRFS)
invert F07TWF (ZTRTRI) F07UWF (ZTPTRI) F07WXF (ZTFTRI)  
Table 8
Routines for complex triangular matrices

4  Functionality Index

Apply iterative refinement to the solution and compute error estimates: 
    after factorizing the matrix of coefficients: 
        complex Hermitian indefinite matrix F07MVF (ZHERFS)
        complex Hermitian indefinite matrix, packed storage F07PVF (ZHPRFS)
        complex Hermitian positive definite band matrix F07HVF (ZPBRFS)
        complex Hermitian positive definite matrix, packed storage F07GVF (ZPPRFS)
        complex Hermitian positive definite tridiagonal matrix F07JVF (ZPTRFS)
        complex symmetric indefinite matrix F07NVF (ZSYRFS)
        complex symmetric indefinite matrix, packed storage F07QVF (ZSPRFS)
        complex tridiagonal matrix F07CVF (ZGTRFS)
        real symmetric indefinite matrix, packed storage  F07PHF (DSPRFS)
        real symmetric positive definite band matrix F07HHF (DPBRFS)
        real symmetric positive definite matrix, packed storage F07GHF (DPPRFS)
        real symmetric positive definite tridiagonal matrix F07JHF (DPTRFS)
        real tridiagonal matrix F07CHF (DGTRFS)
        simple drivers: 
            complex band matrix F07BVF (ZGBRFS)
            complex Hermitian positive definite matrix F07FVF (ZPORFS)
            complex matrix F07AVF (ZGERFS)
            real band matrix F07BHF (DGBRFS)
            real matrix F07AHF (DGERFS)
            real symmetric indefinite matrix F07MHF (DSYRFS)
            real symmetric positive definite matrix F07FHF (DPORFS)
Compute error estimates: 
    complex triangular band matrix F07VVF (ZTBRFS)
    complex triangular matrix F07TVF (ZTRRFS)
    complex triangular matrix, packed storage F07UVF (ZTPRFS)
    real triangular band matrix F07VHF (DTBRFS)
    real triangular matrix F07THF (DTRRFS)
    real triangular matrix, packed storage F07UHF (DTPRFS)
Compute row and column scalings, 
    complex band matrix F07BTF (ZGBEQU)
    complex Hermitian positive definite band matrix F07HTF (ZPBEQU)
    complex Hermitian positive definite matrix F07FTF (ZPOEQU)
    complex Hermitian positive definite matrix, packed storage F07GTF (ZPPEQU)
    complex matrix F07ATF (ZGEEQU)
    real band matrix F07BFF (DGBEQU)
    real matrix F07AFF (DGEEQU)
    real symmetric positive definite band matrix F07HFF (DPBEQU)
    real symmetric positive definite matrix F07FFF (DPOEQU)
    real symmetric positive definite matrix, packed storage F07GFF (DPPEQU)
Condition number estimation: 
    after factorizing the matrix of coefficients: 
        complex band matrix F07BUF (ZGBCON)
        complex Hermitian indefinite matrix F07MUF (ZHECON)
        complex Hermitian indefinite matrix, packed storage F07PUF (ZHPCON)
        complex Hermitian positive definite band matrix F07HUF (ZPBCON)
        complex Hermitian positive definite matrix F07FUF (ZPOCON)
        complex Hermitian positive definite matrix, packed storage F07GUF (ZPPCON)
        complex Hermitian positive definite tridiagonal matrix F07JUF (ZPTCON)
        complex matrix F07AUF (ZGECON)
        complex symmetric indefinite matrix F07NUF (ZSYCON)
        complex symmetric indefinite matrix, packed storage F07QUF (ZSPCON)
        complex tridiagonal matrix F07CUF (ZGTCON)
        real band matrix F07BGF (DGBCON)
        real matrix F07AGF (DGECON)
        real symmetric indefinite matrix F07MGF (DSYCON)
        real symmetric indefinite matrix, packed storage F07PGF (DSPCON)
        real symmetric positive definite band matrix F07HGF (DPBCON)
        real symmetric positive definite matrix F07FGF (DPOCON)
        real symmetric positive definite matrix, packed storage F07GGF (DPPCON)
        real symmetric positive definite tridiagonal matrix F07JGF (DPTCON)
        real tridiagonal matrix F07CGF (DGTCON)
    complex triangular band matrix F07VUF (ZTBCON)
    complex triangular matrix F07TUF (ZTRCON)
    complex triangular matrix, packed storage F07UUF (ZTPCON)
    real triangular band matrix F07VGF (DTBCON)
    real triangular matrix F07TGF (DTRCON)
    real triangular matrix, packed storage F07UGF (DTPCON)
LDLT factorization: 
    complex Hermitian positive definite tridiagonal matrix F07JRF (ZPTTRF)
    real symmetric positive definite tridiagonal matrix F07JDF (DPTTRF)
LLH or UHU factorization: 
    complex Hermitian positive definite band matrix F07HRF (ZPBTRF)
    complex Hermitian positive definite matrix F07FRF (ZPOTRF)
    complex Hermitian positive definite matrix, packed storage F07GRF (ZPPTRF)
    complex Hermitian positive definite matrix, RFP storage F07WRF (ZPFTRF)
    complex Hermitian positive semidefinite matrix F07KRF (ZPSTRF)
LLT or UTU factorization: 
    real symmetric positive definite band matrix F07HDF (DPBTRF)
    real symmetric positive definite matrix F07FDF (DPOTRF)
    real symmetric positive definite matrix, packed storage F07GDF (DPPTRF)
    real symmetric positive definite matrix, RFP storage F07WDF (DPFTRF)
    real symmetric positive semidefinite matrix F07KDF (DPSTRF)
LU factorization: 
    complex band matrix F07BRF (ZGBTRF)
    complex matrix F07ARF (ZGETRF)
    complex tridiagonal matrix F07CRF (ZGTTRF)
    real band matrix F07BDF (DGBTRF)
    real matrix F07ADF (DGETRF)
    real tridiagonal matrix F07CDF (DGTTRF)
Matrix inversion: 
    after factorizing the matrix of coefficients: 
        complex Hermitian indefinite matrix F07MWF (ZHETRI)
        complex Hermitian indefinite matrix, packed storage F07PWF (ZHPTRI)
        complex Hermitian positive definite matrix F07FWF (ZPOTRI)
        complex Hermitian positive definite matrix, packed storage F07GWF (ZPPTRI)
        complex Hermitian positive definite matrix, RFP storage F07WWF (ZPFTRI)
        complex matrix F07AWF (ZGETRI)
        complex symmetric indefinite matrix F07NWF (ZSYTRI)
        complex symmetric indefinite matrix, packed storage F07QWF (ZSPTRI)
        real matrix F07AJF (DGETRI)
        real symmetric indefinite matrix F07MJF (DSYTRI)
        real symmetric indefinite matrix, packed storage F07PJF (DSPTRI)
        real symmetric positive definite matrix F07FJF (DPOTRI)
        real symmetric positive definite matrix, packed storage F07GJF (DPPTRI)
        real symmetric positive definite matrix, RFP storage F07WJF (DPFTRI)
    complex triangular matrix F07TWF (ZTRTRI)
    complex triangular matrix, packed storage F07UWF (ZTPTRI)
    complex triangular matrix, RFP storage, 
        expert driver F07WXF (ZTFTRI)
    real triangular matrix F07TJF (DTRTRI)
    real triangular matrix, packed storage F07UJF (DTPTRI)
    real triangular matrix, RFP storage, 
        expert driver F07WKF (DTFTRI)
PLDLHPH or PUDUHPH factorization: 
    complex Hermitian indefinite matrix F07MRF (ZHETRF)
    complex Hermitian indefinite matrix, packed storage F07PRF (ZHPTRF)
PLDLTPT or PUDUTPT factorization: 
    complex symmetric indefinite matrix F07NRF (ZSYTRF)
    complex symmetric indefinite matrix, packed storage F07QRF (ZSPTRF)
    real symmetric indefinite matrix F07MDF (DSYTRF)
    real symmetric indefinite matrix, packed storage F07PDF (DSPTRF)
Solution of simultaneous linear equations: 
    after factorizing the matrix of coefficients: 
        complex band matrix F07BSF (ZGBTRS)
        complex Hermitian indefinite matrix F07MSF (ZHETRS)
        complex Hermitian indefinite matrix, packed storage F07PSF (ZHPTRS)
        complex Hermitian positive definite band matrix F07HSF (ZPBTRS)
        complex Hermitian positive definite matrix F07FSF (ZPOTRS)
        complex Hermitian positive definite matrix, packed storage F07GSF (ZPPTRS)
        complex Hermitian positive definite matrix, RFP storage F07WSF (ZPFTRS)
        complex Hermitian positive definite tridiagonal  matrix F07JSF (ZPTTRS)
        complex matrix F07ASF (ZGETRS)
        complex symmetric indefinite matrix F07NSF (ZSYTRS)
        complex symmetric indefinite matrix, packed storage F07QSF (ZSPTRS)
        complex tridiagonal matrix F07CSF (ZGTTRS)
        real band matrix F07BEF (DGBTRS)
        real matrix F07AEF (DGETRS)
        real symmetric indefinite matrix F07MEF (DSYTRS)
        real symmetric indefinite matrix, packed storage F07PEF (DSPTRS)
        real symmetric positive definite band matrix F07HEF (DPBTRS)
        real symmetric positive definite matrix F07FEF (DPOTRS)
        real symmetric positive definite matrix, packed storage F07GEF (DPPTRS)
        real symmetric positive definite matrix, RFP storage F07WEF (DPFTRS)
        real symmetric positive definite tridiagonal matrix F07JEF (DPTTRS)
        real tridiagonal matrix F07CEF (DGTTRS)
    expert drivers (with condition and error estimation): 
        complex band matrix F07BPF (ZGBSVX)
        complex Hermitian indefinite matrix F07MPF (ZHESVX)
        complex Hermitian indefinite matrix, packed storage F07PPF (ZHPSVX)
        complex Hermitian positive definite band matrix F07HPF (ZPBSVX)
        complex Hermitian positive definite matrix F07FPF (ZPOSVX)
        complex Hermitian positive definite matrix, packed storage F07GPF (ZPPSVX)
        complex Hermitian positive definite tridiagonal matrix F07JPF (ZPTSVX)
        complex matrix F07APF (ZGESVX)
        complex symmetric indefinite matrix F07NPF (ZSYSVX)
        complex symmetric indefinite matrix, packed storage F07QPF (ZSPSVX)
        complex tridiagonal matrix F07CPF (ZGTSVX)
        real band matrix F07BBF (DGBSVX)
        real matrix F07ABF (DGESVX)
        real symmetric indefinite matrix F07MBF (DSYSVX)
        real symmetric indefinite matrix, packed storage F07PBF (DSPSVX)
        real symmetric positive definite band matrix F07HBF (DPBSVX)
        real symmetric positive definite matrix F07FBF (DPOSVX)
        real symmetric positive definite matrix, packed storage F07GBF (DPPSVX)
        real symmetric positive definite tridiagonal matrix F07JBF (DPTSVX)
        real tridiagonal matrix F07CBF (DGTSVX)
    simple drivers: 
        complex band matrix F07BNF (ZGBSV)
        complex Hermitian indefinite matrix F07MNF (ZHESV)
        complex Hermitian indefinite matrix, packed storage F07PNF (ZHPSV)
        complex Hermitian positive definite band matrix F07HNF (ZPBSV)
        complex Hermitian positive definite matrix F07FNF (ZPOSV)
        complex Hermitian positive definite matrix, packed storage F07GNF (ZPPSV)
        complex Hermitian positive definite matrix, using mixed precision F07FQF (ZCPOSV)
        complex Hermitian positive definite tridiagonal matrix F07JNF (ZPTSV)
        complex matrix F07ANF (ZGESV)
        complex matrix, using mixed precision F07AQF (ZCGESV)
        complex symmetric indefinite matrix F07NNF (ZSYSV)
        complex symmetric indefinite matrix, packed storage F07QNF (ZSPSV)
        complex triangular band matrix F07VSF (ZTBTRS)
        complex triangular matrix F07TSF (ZTRTRS)
        complex triangular matrix, packed storage F07USF (ZTPTRS)
        complex tridiagonal matrix F07CNF (ZGTSV)
        real band matrix F07BAF (DGBSV)
        real matrix F07AAF (DGESV)
        real matrix, using mixed precision F07ACF (DSGESV)
        real symmetric indefinite matrix F07MAF (DSYSV)
        real symmetric indefinite matrix, packed storage F07PAF (DSPSV)
        real symmetric positive definite band matrix F07HAF (DPBSV)
        real symmetric positive definite matrix F07FAF (DPOSV)
        real symmetric positive definite matrix, packed storage F07GAF (DPPSV)
        real symmetric positive definite matrix, using mixed precision F07FCF (DSPOSV)
        real symmetric positive definite tridiagonal matrix F07JAF (DPTSV)
        real triangular band matrix  F07VEF (DTBTRS)
        real triangular matrix F07TEF (DTRTRS)
        real triangular matrix, packed storage F07UEF (DTPTRS)
        real tridiagonal matrix F07CAF (DGTSV)

5  Auxiliary Routines Associated with Library Routine Parameters

None.

6  Routines Withdrawn or Scheduled for Withdrawal

None.

7  References

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

F07 Chapter Contents
F07 Chapter Introduction (PDF version)
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011