NAG Library Chapter Introduction
F08 – Least Squares and Eigenvalue Problems (LAPACK)
1 Scope of the Chapter
This chapter provides routines for the solution of linear least squares problems, eigenvalue problems and singular value problems, as well as associated computations. It provides routines for:
- – solution of linear least squares problems
- – solution of symmetric eigenvalue problems
- – solution of nonsymmetric eigenvalue problems
- – solution of singular value problems
- – solution of generalized symmetric-definite eigenvalue problems
- – solution of generalized nonsymmetric eigenvalue problems
- – solution of generalized singular value problems
- – solution of generalized linear least squares problems
- – matrix factorizations associated with the above problems
- – estimating condition numbers of eigenvalue and eigenvector problems
- – estimating the numerical rank of a matrix
- – solution of the Sylvester matrix equation
Routines are provided for both
real and
complex data.
For a general introduction to the solution of linear least squares problems, you should turn first to
Chapter F04. The decision trees, at the end of
Chapter F04, direct you to the most appropriate routines in
Chapters F04 or
F08.
Chapters F04 and
F08 contain
Black Box (or
driver) routines which enable standard linear least squares problems to be solved by a call to a single routine.
For a general introduction to eigenvalue and singular value problems, you should turn first to
Chapter F02. The decision trees, at the end of
Chapter F02, direct you to the most appropriate routines in
Chapters F02 or
F08.
Chapters F02 and
F08 contain
Black Box (or
driver) routines which enable standard types of problem to be solved by a call to a single routine. Often routines in
Chapter F02 call
Chapter F08 routines to perform the necessary computational tasks.
The routines in this chapter (
Chapter F08) handle only
dense,
band,
tridiagonal and
Hessenberg matrices (not matrices with more specialized structures, or general sparse matrices). The tables in
Section 3 and the decision trees in
Section 4 direct you to the most appropriate routines in
Chapter F08.
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.
It is not expected that you will need to read all of the following sections, but rather you will pick out those sections relevant to your particular problem.
2 Background to the Problems
This section is only a brief introduction to the numerical solution of linear least squares problems, eigenvalue and singular value problems. Consult a standard textbook for a more thorough discussion, for example
Golub and Van Loan (1996).
2.1 Linear Least Squares Problems
The
linear least squares problem is
where
A is an
m by
n matrix,
b is a given
m element vector and
x is an
n element solution vector.
In the most usual case
m≥n and
rankA=n, so that
A has
full rank and in this case the solution to problem
(1) is unique; the problem is also referred to as finding a
least squares solution to an
overdetermined system of linear equations.
When m<n and rankA=m, there are an infinite number of solutions x which exactly satisfy b-Ax=0. In this case it is often useful to find the unique solution x which minimizes x2, and the problem is referred to as finding a minimum norm solution to an underdetermined system of linear equations.
In the general case when we may have rankA<minm,n – in other words, A may be rank-deficient – we seek the minimum norm least squares solution x which minimizes both x2 and b-Ax2.
This chapter (
Chapter F08) contains driver routines to solve these problems with a single call, as well as computational routines that can be combined with routines in
Chapter F07 to solve these linear least squares problems.
Chapter F04 also contains Black Box routines to solve these linear least squares problems in standard cases.
The next two sections discuss the factorizations that can be used in the solution of linear least squares problems.
2.2 Orthogonal Factorizations and Least Squares Problems
A number of routines are provided for factorizing a general rectangular m by n matrix A, as the product of an orthogonal matrix (unitary if complex) and a triangular (or possibly trapezoidal) matrix.
A real matrix
Q is
orthogonal if
QTQ=I; a complex matrix
Q is
unitary if
QHQ=I. Orthogonal or unitary matrices have the important property that they leave the
2-norm of a vector invariant, so that
if
Q is orthogonal or unitary. They usually help to maintain numerical stability because they do not amplify rounding errors.
Orthogonal factorizations are used in the solution of linear least squares problems. They may also be used to perform preliminary steps in the solution of eigenvalue or singular value problems, and are useful tools in the solution of a number of other problems.
2.2.1 QR factorization
The most common, and best known, of the factorizations is the
QR factorization given by
where
R is an
n by
n upper triangular matrix and
Q is an
m by
m orthogonal (or unitary) matrix. If
A is of full rank
n, then
R is nonsingular. It is sometimes convenient to write the factorization as
which reduces to
where
Q1 consists of the first
n columns of
Q, and
Q2 the remaining
m-n columns.
If
m<n,
R is trapezoidal, and the factorization can be written
where
R1 is upper triangular and
R2 is rectangular.
The
QR factorization can be used to solve the linear least squares problem
(1) when
m≥n and
A is of full rank, since
where
and
c1 is an
n element vector. Then
x is the solution of the upper triangular system
The residual vector
r is given by
The residual sum of squares
r22 may be computed without forming
r explicitly, since
2.2.2 LQ factorization
The
LQ factorization is given by
where
L is
m by
m lower triangular,
Q is
n by
n orthogonal (or unitary),
Q1 consists of the first
m rows of
Q, and
Q2 the remaining
n-m rows.
The
LQ factorization of
A is essentially the same as the
QR factorization of
AT (
AH if
A is complex), since
The
LQ factorization may be used to find a minimum norm solution of an underdetermined system of linear equations
Ax=b where
A is
m by
n with
m<n and has rank
m. The solution is given by
2.2.3 QR factorization with column pivoting
To solve a linear least squares problem
(1) when
A is not of full rank, or the rank of
A is in doubt, we can perform either a
QR factorization with column pivoting or a singular value decomposition.
The
QR factorization with column pivoting is given by
where
Q and
R are as before and
P is a (real) permutation matrix, chosen (in general) so that
and moreover, for each
k,
If we put
where
R11 is the leading
k by
k upper triangular sub-matrix of
R then, in exact arithmetic, if
rankA=k, the whole of the sub-matrix
R22 in rows and columns
k+1 to
n would be zero. In numerical computation, the aim must be to determine an index
k, such that the leading sub-matrix
R11 is well-conditioned, and
R22 is negligible, so that
Then
k is the effective rank of
A. See
Golub and Van Loan (1996) for a further discussion of numerical rank determination.
The so-called basic solution to the linear least squares problem
(1) can be obtained from this factorization as
where
c^1 consists of just the first
k elements of
c=QTb.
2.2.4 Complete orthogonal factorization
The
Q
R
factorization with column pivoting does not enable us to compute a
minimum norm solution to a rank-deficient linear least squares problem, unless
R12
=
0
. However, by applying for further orthogonal (or unitary) transformations from the right to the upper trapezoidal matrix
R11
R12
,
R12
can be eliminated:
This gives the
complete orthogonal factorization
from which the minimum norm solution can be obtained as
2.2.5 Other factorizations
The
QL and
RQ factorizations are given by
and
The factorizations are less commonly used than either the QR or LQ factorizations described above, but have applications in, for example, the computation of generalized QR factorizations.
2.3 The Singular Value Decomposition
The
singular value decomposition (SVD) of an
m by
n matrix
A is given by
where
U and
V are orthogonal (unitary) and
Σ is an
m by
n diagonal matrix with real diagonal elements,
σi, such that
The
σi are the
singular values of
A and the first
minm,n columns of
U and
V are the
left and
right singular vectors of
A. The singular values and singular vectors satisfy
where
ui and
vi are the
ith columns of
U and
V respectively.
The computation proceeds in the following stages.
- The matrix A is reduced to bidiagonal form A=U1BV1T if A is real (A=U1BV1H if A is complex), where U1 and V1 are orthogonal (unitary if A is complex), and B is real and upper bidiagonal when m≥n and lower bidiagonal when m<n, so that B is nonzero only on the main diagonal and either on the first superdiagonal (if m≥n) or the first subdiagonal (if m<n).
- The SVD of the bidiagonal matrix B is computed as B=U2Σ
V2T
, where U2 and V2 are orthogonal and Σ is diagonal as described above. The singular vectors of A are then U=U1U2 and V=V1V2.
If m≫n, it may be more efficient to first perform a QR factorization of A, and then compute the SVD of the n by n matrix R, since if A=QR and R=UΣVT, then the SVD of A is given by A=QUΣVT.
Similarly, if m≪n, it may be more efficient to first perform an LQ factorization of A.
This chapter supports two primary algorithms for computing the SVD of a bidiagonal matrix. They are:
| (i) |
the divide and conquer algorithm; |
| (ii) |
the QR algorithm. |
The divide and conquer algorithm is much faster than the QR algorithm if singular vectors of large matrices are required.
2.4 The Singular Value Decomposition and Least Squares Problems
The SVD may be used to find a minimum norm solution to a (possibly) rank-deficient linear least squares problem
(1). The effective rank,
k, of
A can be determined as the number of singular values which exceed a suitable threshold. Let
Σ^ be the leading
k by
k sub-matrix of
Σ, and
V^ be the matrix consisting of the first
k columns of
V. Then the solution is given by
where
c^1 consists of the first
k elements of
c=UTb=
U2TU1T
b.
2.5 Generalized Linear Least Squares Problems
The simple type of linear least squares problem described in
Section 2.1 can be generalized in various ways.
- Linear least squares problems with equality constraints:
where A is m by n and B is p by n, with p≤n≤m+p. The equations Bx=d may be regarded as a set of equality constraints on the problem of minimizing S. Alternatively the problem may be regarded as solving an overdetermined system of equations
where some of the equations (those involving B) are to be solved exactly, and the others (those involving A) are to be solved in a least squares sense. The problem has a unique solution on the assumptions that B has full row rank p and the matrix
A
B
has full column rank n. (For linear least squares problems with inequality constraints, refer to Chapter E04.)
- General Gauss–Markov linear model problems:
where A is m by n and B is m by p, with n≤m≤n+p. When B=I, the problem reduces to an ordinary linear least squares problem. When B is square and nonsingular, it is equivalent to a weighted linear least squares problem:
The problem has a unique solution on the assumptions that A has full column rank n, and the matrix A,B has full row rank m. Unless B is diagonal, for numerical stability it is generally preferable to solve a weighted linear least squares problem as a general Gauss–Markov linear model problem.
2.6 Generalized Orthogonal Factorization and Generalized Linear Least Squares Problems
2.6.1 Generalized QR Factorization
The
generalized QR (GQR) factorization of an
n by
m matrix
A and an
n by
p matrix
B is given by the pair of factorizations
where
Q and
Z are respectively
n by
n and
p by
p orthogonal matrices (or unitary matrices if
A and
B are complex).
R has the form
or
where
R11
is upper triangular.
T has the form
or
where
T12
or
T21
is upper triangular.
Note that if
B is square and nonsingular, the GQR factorization of
A and
B implicitly gives the
QR factorization of the matrix
B-1
A
:
without explicitly computing the matrix inverse
B-1
or the product
B-1
A
.
The GQR factorization can be used to solve the general (Gauss–Markov) linear model problem (GLM) (see
Section 2.5, but note that
A and
B are dimensioned differently there as
m by
n and
p by
n respectively). Using the GQR factorization of
A and
B, we rewrite the equation
d
=
Ax
+
By
as
We partition this as
where
The GLM problem is solved by setting
from which we obtain the desired solutions
2.6.2 Generalized RQ Factorization
The
generalized RQ (GRQ) factorization of an
m by
n matrix
A and a
p by
n matrix
B is given by the pair of factorizations
where
Q and
Z are respectively
n by
n and
p by
p orthogonal matrices (or unitary matrices if
A and
B are complex).
R has the form
or
where
R12
or
R21
is upper triangular.
T has the form
or
where
T11
is upper triangular.
Note that if
B is square and nonsingular, the GRQ factorization of
A and
B implicitly gives the
RQ factorization of the matrix
AB-1
:
without explicitly computing the matrix
B-1
or the product
AB-1
.
The GRQ factorization can be used to solve the linear equality-constrained least squares problem (LSE) (see
Section 2.5). We use the GRQ factorization of
B and
A (note that
B and
A have swapped roles), written as
We write the linear equality constraints
Bx=d
as
which we partition as:
Therefore
x2
is the solution of the upper triangular system
We partition this expression as:
where
c1
c2
≡
ZTc
.
To solve the LSE problem, we set
which gives
x1
as the solution of the upper triangular system
Finally, the desired solution is given by
2.6.3 Generalized Singular Value Decomposition (GSVD)
The
generalized (or quotient) singular value decomposition of an
m by
n matrix
A and a
p by
n matrix
B is given by the pair of factorizations
The matrices in these factorizations have the following properties:
| – |
U is m by m, V is p by p, Q is n by n, and all three matrices are orthogonal. If A and B are complex, these matrices are unitary instead of orthogonal, and
QT
should be replaced by
QH
in the pair of factorizations. |
| – |
R is r by r, upper triangular and nonsingular.
0,R
is r by n (in other words, the 0 is an r by n-r zero matrix). The integer r is the rank of
A
B
, and satisfies r≤n. |
| – |
Σ1
is m by r,
Σ2
is p by r, both are real, non-negative and diagonal, and
Σ1T
Σ1
+
Σ2T
Σ2
=
I
. Write
Σ1T
Σ1
=
diagα12,…,αr2
and
Σ2T
Σ2
=
diagβ12,…,βr2
, where
αi
and
βi
lie in the interval from 0 to 1. The ratios
α1
/
β1
,…,
αr
/
βr
are called the generalized singular values of the pair A, B. If
βi
=
0
, then the generalized singular value
αi
/
βi
is infinite. |
Σ1
and
Σ2
have the following detailed structures, depending on whether
m-r≥0
or
m-r<0
. In the first case,
m-r≥0
, then
Here l is the rank of B,
k=r-l
, C and S are diagonal matrices satisfying
C2
+
S2
=
I
, and S is nonsingular. We may also identify
α1
=
⋯
=
αk
=
1
,
αk+i
=
cii
, for
i=1,2,…,
l,
β1
=
⋯
=
βk
=
0
, and
βk+i
=
sii
, for
i=1,2,…,
l
. Thus, the first k generalized singular values
α1
/
β1
,…,
αk
/
βk
are infinite, and the remaining l generalized singular values are finite.
In the second case, when
m-r<0
,
and
Again, l is the rank of B,
k=r-l
, C and S are diagonal matrices satisfying
C2
+
S2
=
I
, and S is nonsingular, and we may identify
α1
=
⋯
=
αk
=
1
,
αk+i
=
cii
, for
i=1,2,…,
m-k
,
αm+1
=
⋯
=
αr
=
0
,
β1
=
⋯
=
βk
=
0
,
βk+i
=
sii
, for
i=1,2,…,
m-k
and
βm+1
=
⋯
=
βr
=
1
. Thus, the first k generalized singular values
α1
/
β1
,…,
αk
/
βk
are infinite, and the remaining l generalized singular values are finite.
Here are some important special case of the generalized singular value decomposition. First, if
B is square and nonsingular, then
r=n and the generalized singular value decomposition of
A and
B is equivalent to the singular value decomposition of
AB-1
, where the singular values of
AB-1
are equal to the generalized singular values of the pair
A,
B:
Second, if the columns of
AT
BT
T
are orthonormal, then
r=n,
R=I and the generalized singular value decomposition of
A and
B is equivalent to the CS (Cosine–Sine) decomposition of
AT
BT
T
:
Third, the generalized eigenvalues and eigenvectors of
ATA
-
λ
BTB
can be expressed in terms of the generalized singular value decomposition: Let
Therefore, the columns of
X are the eigenvectors of
ATA
-
λ
BTB
, and ‘nontrivial’ eigenvalues are the squares of the generalized singular values (see also
Section 2.8). ‘Trivial’ eigenvalues are those corresponding to the leading
n-r columns of
X, which span the common null space of
ATA
and
BTB
. The ‘trivial eigenvalues’ are not well defined.
2.7 Symmetric Eigenvalue Problems
The
symmetric eigenvalue problem is to find the
eigenvalues,
λ, and corresponding
eigenvectors,
z≠0, such that
For the
Hermitian eigenvalue problem we have
For both problems the eigenvalues
λ are real.
When all eigenvalues and eigenvectors have been computed, we write
where
Λ is a diagonal matrix whose diagonal elements are the eigenvalues, and
Z is an orthogonal (or unitary) matrix whose columns are the eigenvectors. This is the classical
spectral factorization of
A.
The basic task of the symmetric eigenproblem routines is to compute values of
λ and, optionally, corresponding vectors
z for a given matrix
A. This computation proceeds in the following stages.
- The real symmetric or complex Hermitian matrix A is reduced to real tridiagonal form
T. If A is real symmetric this decomposition is A=QTQT with Q orthogonal and T symmetric tridiagonal. If A is complex Hermitian, the decomposition is A=QTQH with Q unitary and T, as before, real symmetric tridiagonal.
- Eigenvalues and eigenvectors of the real symmetric tridiagonal matrix T are computed. If all eigenvalues and eigenvectors are computed, this is equivalent to factorizing T as T=SΛST, where S is orthogonal and Λ is diagonal. The diagonal entries of Λ are the eigenvalues of T, which are also the eigenvalues of A, and the columns of S are the eigenvectors of T; the eigenvectors of A are the columns of Z=QS, so that A=ZΛZT (ZΛZH when A is complex Hermitian).
This chapter supports four primary algorithms for computing eigenvalues and eigenvectors of real symmetric matrices and complex Hermitian matrices. They are:
| (i) |
the divide-and-conquer algorithm; |
| (ii) |
the QR algorithm; |
| (iii) |
bisection followed by inverse iteration; |
| (iv) |
the Relatively Robust Representation (RRR). |
The divide-and-conquer algorithm is generally more efficient than the traditional
QR algorithm for computing all eigenvalues and eigenvectors, but the RRR algorithm tends to be fastest of all. For further information and references see
Anderson et al. (1999).
2.8 Generalized Symmetric-definite Eigenvalue Problems
This section is concerned with the solution of the generalized eigenvalue problems Az=λBz, ABz=λz, and BAz=λz, where A and B are real symmetric or complex Hermitian and B is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of B as either B=LLT or B=UTU (LLH or UHU in the Hermitian case).
With
B=LLT, we have
Hence the eigenvalues of
Az=λBz are those of
Cy=λy, where
C is the symmetric matrix
C=L-1AL-T and
y=LTz. In the complex case
C is Hermitian with
C=L-1AL-H and
y=LHz.
Table 1 summarizes how each of the three types of problem may be reduced to standard form
Cy=λy, and how the eigenvectors
z of the original problem may be recovered from the eigenvectors
y of the reduced problem. The table applies to real problems; for complex problems, transposed matrices must be replaced by conjugate-transposes.
| |
Type of problem |
Factorization of B |
Reduction |
Recovery of eigenvectors |
| 1. |
Az=λBz |
B=LLT,
B=UTU |
C=L-1AL-T,
C=U-TAU-1 |
z=L-Ty,
z=U-1y |
| 2. |
ABz=λz |
B=LLT,
B=UTU |
C=LTAL,
C=UAUT |
z=L-Ty,
z=U-1y |
| 3. |
BAz=λz |
B=LLT,
B=UTU |
C=LTAL,
C=UAUT |
z=Ly,
z=UTy |
Table 1
Reduction of generalized symmetric-definite eigenproblems to standard problems
When the generalized symmetric-definite problem has been reduced to the corresponding standard problem
Cy=λy, this may then be solved using the routines described in the previous section. No special routines are needed to recover the eigenvectors
z of the generalized problem from the eigenvectors
y of the standard problem, because these computations are simple applications of Level 2 or Level 3 BLAS
(see
Chapter F06).
2.9 Packed Storage for Symmetric Matrices
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 either 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; that is, the storage is almost halved.
This storage format is referred to as
packed storage; it is described in
Section 3.3.2 in the F07 Chapter Introduction.
Routines designed for packed storage are usually less efficient, especially on high-performance computers, so there is a trade-off between storage and efficiency.
2.10 Band Matrices
A
band matrix is one whose elements are confined to a relatively small number of subdiagonals or superdiagonals on either side of the main diagonal. Algorithms can take advantage of bandedness to reduce the amount of work and storage required. The storage scheme for band matrices is described in
Section 3.3.4 in the F07 Chapter Introduction.
If the problem is the generalized symmetric definite eigenvalue problem
Az=λBz and the matrices
A and
B are additionally banded, the matrix
C as defined in
Section 2.8 is, in general, full. We can reduce the problem to a banded standard problem by modifying the definition of
C thus:
where
Q is an orthogonal matrix chosen to ensure that
C has bandwidth no greater than that of
A.
A further refinement is possible when
A and
B are banded, which halves the amount of work required to form
C. Instead of the standard Cholesky factorization of
B as
UTU or
LLT, we use a
split Cholesky factorization
B=STS, where
with
U11 upper triangular and
L22 lower triangular of order approximately
n/2;
S has the same bandwidth as
B.
2.11 Nonsymmetric Eigenvalue Problems
The
nonsymmetric eigenvalue problem is to find the
eigenvalues,
λ, and corresponding
eigenvectors,
v≠0, such that
More precisely, a vector
v as just defined is called a
right eigenvector of
A, and a vector
u≠0 satisfying
is called a
left eigenvector of
A.
A real matrix A may have complex eigenvalues, occurring as complex conjugate pairs.
This problem can be solved via the
Schur factorization of
A, defined in the real case as
where
Z is an orthogonal matrix and
T is an upper quasi-triangular matrix with
1 by
1 and
2 by
2 diagonal blocks, the
2 by
2 blocks corresponding to complex conjugate pairs of eigenvalues of
A. In the complex case, the Schur factorization is
where
Z is unitary and
T is a complex upper triangular matrix.
The columns of Z are called the Schur vectors. For each k (1≤k≤n), the first k columns of Z form an orthonormal basis for the invariant subspace corresponding to the first k eigenvalues on the diagonal of T. Because this basis is orthonormal, it is preferable in many applications to compute Schur vectors rather than eigenvectors. It is possible to order the Schur factorization so that any desired set of k eigenvalues occupy the k leading positions on the diagonal of T.
The two basic tasks of the nonsymmetric eigenvalue routines are to compute, for a given matrix A, all n values of λ and, if desired, their associated right eigenvectors v and/or left eigenvectors u, and the Schur factorization.
These two basic tasks can be performed in the following stages.
- A general matrix A is reduced to upper Hessenberg form
H which is zero below the first subdiagonal. The reduction may be written A=QHQT with Q orthogonal if A is real, or A=QHQH with Q unitary if A is complex.
- The upper Hessenberg matrix H is reduced to Schur form T, giving the Schur factorization H=STST (for H real) or H=STSH (for H complex). The matrix S (the Schur vectors of H) may optionally be computed as well. Alternatively S may be postmultiplied into the matrix Q determined in stage 1, to give the matrix Z=QS, the Schur vectors of A. The eigenvalues are obtained from the diagonal elements or diagonal blocks of T.
- Given the eigenvalues, the eigenvectors may be computed in two different ways. Inverse iteration can be performed on H to compute the eigenvectors of H, and then the eigenvectors can be multiplied by the matrix Q in order to transform them to eigenvectors of A. Alternatively the eigenvectors of T can be computed, and optionally transformed to those of H or A if the matrix S or Z is supplied.
The accuracy with which eigenvalues can be obtained can often be improved by
balancing a matrix. This is discussed further in
Section 2.14.6 below.
2.12 Generalized Nonsymmetric Eigenvalue Problem
The
generalized nonsymmetric eigenvalue problem is to find the eigenvalues,
λ, and corresponding
eigenvectors,
v≠0, such that
More precisely, a vector
v as just defined is called a
right eigenvector of the matrix pair
A,B, and a vector
u≠0 satisfying
is called a
left eigenvector of the matrix pair
A,B.
If
B is singular then the problem has one or more
infinite eigenvalues
λ=∞, corresponding to
Bv=0. Note that if
A is nonsingular, then the equivalent problem
μAv=Bv is perfectly well defined and an infinite eigenvalue corresponds to
μ=0. To deal with both finite (including zero) and infinite eigenvalues, the routines in this chapter do not compute
λ explicitly, but rather return a pair of numbers
α,β such that if
β≠0
and if
α≠0 and
β=0 then
λ=∞.
β is always returned as real and non-negative. Of course, computationally an infinite eigenvalue may correspond to a small
β rather than an exact zero.
For a given pair
A,B the set of all the matrices of the form
A-λB is called a matrix
pencil and
λ and
v are said to be an eigenvalue and eigenvector of the pencil
A-λB. If
A and
B are both singular and share a common null space then
so that the pencil
A-λB is
singular for all
λ. In other words any
λ can be regarded as an eigenvalue. In exact arithmetic a singular pencil will have
α=β=0 for some
α,β. Computationally if some pair
α,β is small then the pencil is singular, or nearly singular, and no reliance can be placed on any of the computed eigenvalues. Singular pencils can also manifest themselves in other ways; see, in particular, Sections 2.3.5.2 and 4.11.1.4 of
Anderson et al. (1999) for further details.
The generalized eigenvalue problem can be solved via the
generalized Schur factorization of the pair
A,B defined in the real case as
where
Q and
Z are orthogonal,
T is upper triangular with non-negative diagonal elements and
S is upper quasi-triangular with
1 by
1 and
2 by
2 diagonal blocks, the
2 by
2 blocks corresponding to complex conjugate pairs of eigenvalues. In the complex case, the generalized Schur factorization is
where
Q and
Z are unitary and
S and
T are upper triangular, with
T having real non-negative diagonal elements. The columns of
Q and
Z are called respectively the
left and
right generalized Schur vectors and span pairs of
deflating subspaces of
A and
B, which are a generalization of invariant subspaces.
It is possible to order the generalized Schur factorization so that any desired set of k eigenvalues correspond to the k leading positions on the diagonals of the pair S,T.
The two basic tasks of the generalized nonsymmetric eigenvalue routines are to compute, for a given pair A,B, all n values of λ and, if desired, their associated right eigenvectors v and/or left eigenvectors u, and the generalized Schur factorization.
These two basic tasks can be performed in the following stages.
- The matrix pair A,B is reduced to generalized upper Hessenberg form H,R,
where H is upper Hessenberg (zero below the first subdiagonal) and R is upper triangular. The reduction may be written as A=Q1HZ1T,
B=Q1RZ1T in the real case with Q1 and Z1 orthogonal, and
A=Q1H
Z1H
,
B=Q1R
Z1H
in the complex case with Q1 and Z1 unitary.
- The generalized upper Hessenberg form H,R is reduced to the generalized
Schur form S,T using the generalized Schur factorization H=Q2S
Z2T, R=Q2T
Z2T in the real case with Q2 and Z2 orthogonal, and H=Q2
SZ2H,
R=Q2T
Z2H in the complex case. The generalized Schur vectors of A,B are given by Q=Q1Q2, Z=Z1Z2. The eigenvalues are obtained from the diagonal elements (or blocks) of the pair S,T.
- Given the eigenvalues, the eigenvectors of the pair S,T can be computed, and optionally transformed to those of H,R or A,B.
The accuracy with which eigenvalues can be obtained can often be improved by
balancing a matrix pair. This is discussed further in
Section 2.14.8 below.
2.13 The Sylvester Equation and the Generalized Sylvester Equation
The Sylvester equation is a matrix equation of the form
where
A,
B, and
C are given matrices with
A being
m by
m,
B an
n by
n matrix and
C, and the solution matrix
X,
m by
n matrices. The solution of a special case of this equation occurs in the computation of the condition number for an invariant subspace, but a combination of routines in this chapter allows the solution of the general Sylvester equation.
Routines are also provided for solving a special case of the generalized Sylvester equations
where
A,D,
B,E and
C,F are given matrix pairs, and
R and
L are the solution matrices.
2.14 Error and Perturbation Bounds and Condition Numbers
In this section we discuss the effects of rounding errors in the solution process and the effects of uncertainties in the data, on the solution to the problem. A number of the routines in this chapter return information, such as condition numbers, that allow these effects to be assessed. First we discuss some notation used in the error bounds of later sections.
The bounds usually contain the factor
pn (or
pm,n), which grows as a function of the matrix dimension
n (or matrix dimensions
m and
n). It measures how errors can grow as a function of the matrix dimension, and represents a potentially different function for each problem. In practice, it usually grows just linearly;
pn≤10n is often true, although generally only much weaker bounds can be actually proved. We normally describe
pn as a ‘modestly growing’ function of
n. For detailed derivations of various
pn, see
Golub and Van Loan (1996) and
Wilkinson (1965).
For linear equation (see
Chapter F07) and least squares solvers, we consider bounds on the relative error
x-x^/x in the computed solution
x^, where
x is the true solution. For eigenvalue problems we consider bounds on the error
λi-λ^i in the
ith computed eigenvalue
λ^i, where
λi is the true
ith eigenvalue. For singular value problems we similarly consider bounds
σi-σ^i.
Bounding the error in computed eigenvectors and singular vectors
v^i is more subtle because these vectors are not unique: even though we restrict
v^i2=1 and
vi2=1, we may still multiply them by arbitrary constants of absolute value
1. So to avoid ambiguity we bound the
angular difference between
v^i and the true vector
vi, so that
Here
arccosθ is in the standard range:
0≤arccosθ<π. When
θvi,v^i is small, we can choose a constant
α with absolute value
1 so that
αvi-v^i2≈θvi,v^i.
In addition to bounds for individual eigenvectors, bounds can be obtained for the spaces spanned by collections of eigenvectors. These may be much more accurately determined than the individual eigenvectors which span them. These spaces are called
invariant subspaces in the case of eigenvectors, because if
v is any vector in the space,
Av is also in the space, where
A is the matrix. Again, we will use angle to measure the difference between a computed space
S^ and the true space
S:
θS,S^ may be computed as follows. Let
S be a matrix whose columns are orthonormal and
spanS. Similarly let
S^ be an orthonormal matrix with columns spanning
S^. Then
Finally, we remark on the accuracy of the bounds when they are large. Relative errors like
x^-x/x and angular errors like
θv^i,vi are only of interest when they are much less than
1. Some stated bounds are not strictly true when they are close to
1, but rigorous bounds are much more complicated and supply little extra information in the interesting case of small errors. These bounds are indicated by using the symbol
≲, or ‘approximately less than’, instead of the usual
≤. Thus, when these bounds are close to 1 or greater, they indicate that the computed answer may have no significant digits at all, but do not otherwise bound the error.
A number of routines in this chapter return error estimates and/or condition number estimates directly. In other cases
Anderson et al. (1999) gives code fragments to illustrate the computation of these estimates, and a number of the
Chapter F08 example programs, for the driver routines, implement these code fragments.
2.14.1 Least squares problems
The conventional error analysis of linear least squares problems goes as follows. The problem is to find the x minimizing Ax-b2. Let x^ be the solution computed using one of the methods described above. We discuss the most common case, where A is overdetermined (i.e., has more rows than columns) and has full rank.
Then the computed solution
x^ has a small normwise backward error. In other words
x^ minimizes
A+Ex^-b+f2, where
and
pn is a modestly growing function of
n and
ε is the
machine precision. Let
κ2A=σmaxA/σminA,
ρ=Ax-b2, and
sinθ=ρ/b2. Then if
pnε is small enough, the error
x^-x is bounded by
If
A is rank-deficient, the problem can be
regularized by treating all singular values less than a user-specified threshold as exactly zero. See
Golub and Van Loan (1996) for error bounds in this case, as well as for the underdetermined case.
The solution of the overdetermined, full-rank problem may also be characterized as the solution of the linear system of equations
By solving this linear system (see
Chapter F07) component-wise error bounds can also be obtained
Arioli et al. (1989).
2.14.2 The singular value decomposition
The usual error analysis of the SVD algorithm is as follows (see
Golub and Van Loan (1996)).
The computed SVD,
U^Σ^V^T, is nearly the exact SVD of
A+E, i.e.,
A+E=U^+δU^Σ^V^+δV^ is the true SVD, so that
U^+δU^ and
V^+δV^ are both orthogonal, where
E2/A2≤pm,nε,
δU^≤pm,nε, and
δV^≤pm,nε. Here
pm,n is a modestly growing function of
m and
n and
ε is the
machine precision. Each computed singular value
σ^i differs from the true
σi by an amount satisfying the bound
Thus large singular values (those near
σ1) are computed to high relative accuracy and small ones may not be.
The angular difference between the computed left singular vector
u^i and the true
ui satisfies the approximate bound
where
is the
absolute gap between
σi and the nearest other singular value. Thus, if
σi is close to other singular values, its corresponding singular vector
ui may be inaccurate. The same bound applies to the computed right singular vector
v^i and the true vector
vi. The gaps may be easily obtained from the computed singular values.
Let
S^ be the space spanned by a collection of computed left singular vectors
u^i,i∈I, where
I is a subset of the integers from
1 to
n. Let
S be the corresponding true space. Then
where
is the absolute gap between the singular values in
I and the nearest other singular value. Thus, a cluster of close singular values which is far away from any other singular value may have a well determined space
S^ even if its individual singular vectors are ill-conditioned. The same bound applies to a set of right singular vectors
v^i,i∈I.
In the special case of bidiagonal matrices, the singular values and singular vectors may be computed much more accurately (see
Demmel and Kahan (1990)). A bidiagonal matrix
B has nonzero entries only on the main diagonal and the diagonal immediately above it (or immediately below it). Reduction of a dense matrix to bidiagonal form
B can introduce additional errors, so the following bounds for the bidiagonal case do not apply to the dense case.
Using the routines in this chapter, each computed singular value of a bidiagonal matrix is accurate to nearly full relative accuracy, no matter how tiny it is, so that
The computed left singular vector
u^i has an angular error at most about
where
is the
relative gap between
σi and the nearest other singular value. The same bound applies to the right singular vector
v^i and
vi. Since the relative gap may be much larger than the absolute gap, this error bound may be much smaller than the previous one. The relative gaps may be easily obtained from the computed singular values.
2.14.3 The symmetric eigenproblem
The usual error analysis of the symmetric eigenproblem is as follows (see
Parlett (1998)).
The computed eigendecomposition
Z^Λ^Z^T is nearly the exact eigendecomposition of
A+E, i.e.,
A+E=Z^+δZ^Λ^Z^+δZ^T is the true eigendecomposition so that
Z^+δZ^ is orthogonal, where
E2/A2≤pnε and
δZ^2≤pnε and
pn is a modestly growing function of
n and
ε is the
machine precision. Each computed eigenvalue
λ^i differs from the true
λi by an amount satisfying the bound
Thus large eigenvalues (those near
max
i
λi
=
A2
) are computed to high relative accuracy and small ones may not be.
The angular difference between the computed unit eigenvector
z^i and the true
zi satisfies the approximate bound
if
pnε is small enough, where
is the
absolute gap between
λi and the nearest other eigenvalue. Thus, if
λi is close to other eigenvalues, its corresponding eigenvector
zi may be inaccurate. The gaps may be easily obtained from the computed eigenvalues.
Let
S^ be the invariant subspace spanned by a collection of eigenvectors
z^i,i∈I, where
I is a subset of the integers from
1 to
n. Let
S be the corresponding true subspace. Then
where
is the absolute gap between the eigenvalues in
I and the nearest other eigenvalue. Thus, a cluster of close eigenvalues which is far away from any other eigenvalue may have a well determined invariant subspace
S^ even if its individual eigenvectors are ill-conditioned.
In the special case of a real symmetric tridiagonal matrix
T, routines in this chapter can compute the eigenvalues and eigenvectors much more accurately. See
Anderson et al. (1999) for further details.
2.14.4 The generalized symmetric-definite eigenproblem
The three types of problem to be considered are
A-λB,
AB-λI and
BA-λI. In each case
A and
B are real symmetric (or complex Hermitian) and
B is positive definite. We consider each case in turn, assuming that routines in this chapter are used to transform the generalized problem to the standard symmetric problem, followed by the solution of the symmetric problem. In all cases
is the
absolute gap between
λi and the nearest other eigenvalue.
- A-λB. The computed eigenvalues λ^i can differ from the true eigenvalues λi by an amount
The angular difference between the computed eigenvector z^i and the true eigenvector zi is
- AB-λI or BA-λI. The computed eigenvalues λ^i can differ from the true eigenvalues λi by an amount
The angular difference between the computed eigenvector z^i and the true eigenvector zi is
These error bounds are large when
B is ill-conditioned with respect to inversion (
κ2B is large). It is often the case that the eigenvalues and eigenvectors are much better conditioned than indicated here. One way to get tighter bounds is effective when the diagonal entries of
B differ widely in magnitude, as for example with a
graded matrix.
- A-λB. Let
D =
diag
b11-1/2
,…,
b
nn
-1/2
be a diagonal matrix. Then replace B by DBD and A by DAD in the above bounds.
- AB-λI or BA-λI. Let D=diagb11-1/2,…,bnn
-1/2 be a diagonal matrix. Then replace B by DBD and A by D-1AD-1 in the above bounds.
Further details can be found in
Anderson et al. (1999).
2.14.5 The nonsymmetric eigenproblem
The nonsymmetric eigenvalue problem is more complicated than the symmetric eigenvalue problem. In this section, we just summarize the bounds. Further details can be found in
Anderson et al. (1999).
We let λ^i be the ith computed eigenvalue and λi the ith true eigenvalue. Let v^i be the corresponding computed right eigenvector, and vi the true right eigenvector (so Avi=λi
vi). If I is a subset of the integers from 1 to n, we let λI denote the average of the selected eigenvalues:
λI=∑i∈I
λi/
∑i∈I1, and similarly for λ^I. We also let SI denote the subspace spanned by vi,i∈I; it is called a right invariant subspace because if v is any vector in SI then Av is also in SI. S^I is the corresponding computed subspace.
The algorithms for the nonsymmetric eigenproblem are normwise backward stable: they compute the exact eigenvalues, eigenvectors and invariant subspaces of slightly perturbed matrices
A+E
E
, where
E
≤
p
n
ε
A
. Some of the bounds are stated in terms of E2 and others in terms of EF; one may use pnε for either quantity.
Routines are provided so that, for each (
λ^i,v^i) pair the two values
si and
sepi, or for a selected subset
I of eigenvalues the values
sI and
sepI can be obtained, for which the error bounds in
Table 2 are true for sufficiently small
E, (which is why they are called asymptotic):
| Simple eigenvalue |
λ^i-λi≲E2/si |
| Eigenvalue cluster |
λ^I-λI≲E2/sI |
| Eigenvector |
θv^i,vi≲EF/sepi |
| Invariant subspace |
θS^I,SI≲EF/sepI |
Table 2
Asymptotic error bounds for the nonsymmetric eigenproblem
If the problem is ill-conditioned, the asymptotic bounds may only hold for extremely small
E. The global error bounds of
Table 3 are guaranteed to hold for all
EF<s×sep/4:
| Simple eigenvalue |
λ^i-λi≤nE2/si |
Holds for all E |
| Eigenvalue cluster |
λ^I-λI≤2E2/sI |
Requires EF<sI×sepI/4 |
| Eigenvector |
θv^i,vi≤arctan2EF/sepi-4EF/si |
Requires EF<si×sepi/4 |
| Invariant subspace |
θS^I,SI≤arctan2EF/sepI-4EF/sI |
Requires EF<sI×sepI/4 |
Table 3
Global error bounds for the nonsymmetric eigenproblem
2.14.6 Balancing and condition for the nonsymmetric eigenproblem
There are two preprocessing steps one may perform on a matrix
A in order to make its eigenproblem easier. The first is
permutation, or reordering the rows and columns to make
A more nearly upper triangular (closer to Schur form):
A′=PAPT, where
P is a permutation matrix. If
A′ is permutable to upper triangular form (or close to it), then no floating point operations (or very few) are needed to reduce it to Schur form. The second is
scaling by a diagonal matrix
D to make the rows and columns of
A′ more nearly equal in norm:
A′′=DA′D-1. Scaling can make the matrix norm smaller with respect to the eigenvalues, and so possibly reduce the inaccuracy contributed by roundoff (see Chapter 11 of
Wilkinson and Reinsch (1971)). We refer to these two operations as
balancing.
Permuting has no effect on the condition numbers or their interpretation as described previously. Scaling, however, does change their interpretation and further details can be found in
Anderson et al. (1999).
2.14.7 The generalized nonsymmetric eigenvalue problem
The algorithms for the generalized nonsymmetric eigenvalue problem are normwise backward stable: they compute the exact eigenvalues (as the pairs
α,β), eigenvectors and deflating subspaces of slightly perturbed pairs
A+E,B+F, where
Asymptotic and global error bounds can be obtained, which are generalizations of those given in
Tables 2 and
3. See Section 4.11 of
Anderson et al. (1999) for details. Routines are provided to compute estimates of reciprocal conditions numbers for eigenvalues and eigenspaces.
2.14.8 Balancing the generalized eigenvalue problem
As with the standard nonsymmetric eigenvalue problem, there are two preprocessing steps one may perform on a matrix pair
A,B in order to make its eigenproblem easier; permutation and scaling, which together are referred to as balancing, as indicated in the following two steps.
- The balancing routine first attempts to permute A and B to block upper triangular form by a similarity transformation:
where P is a permutation matrix, F11, F33, G11 and G33 are upper triangular. Then the diagonal elements of the matrix F11,G11 and G33,H33 are generalized eigenvalues of A,B. The rest of the generalized eigenvalues are given by the matrix pair F22,G22. Subsequent operations to compute the eigenvalues of A,B need only be applied to the matrix F22,G22; this can save a significant amount of work if F22,G22 is smaller than the original matrix pair A,B. If no suitable permutation exists (as is often the case), then there is no gain in efficiency or accuracy.
- The balancing routine applies a diagonal similarity transformation to F,G, to make the rows and columns of F22,G22 as close as possible in the norm:
This transformation usually improves the accuracy of computed generalized eigenvalues and eigenvectors. However, there are exceptional occasions when this transformation increases the norm of the pencil; in this case accuracy could be lower with diagonal balancing.
See
Anderson et al. (1999) for further details.
2.14.9 Other problems
Error bounds for other problems such as the generalized linear least squares problem and generalized singular value decomposition can be found in
Anderson et al. (1999).
2.15 Block Partitioned Algorithms
A number 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 much 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. In the case of the
QR algorithm for reducing an upper Hessenberg matrix to Schur form, a multishift strategy is used in order to improve performance. See
Golub and Van Loan (1996) or
Anderson et al. (1999) for more about block partitioned algorithms and the multishift strategy.
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 more conventional machines there is often 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.
3 Recommendations on Choice and Use of Available Routines
3.1 Available Routines
The tables in the following sub-sections show the routines which are provided for performing different computations on different types of matrices. Each entry in the table gives the NAG routine
name
and the LAPACK double precision name
(see
Section 3.2).
Black box (or driver) routines are provided for the solution of most problems. In a number of cases there are simple drivers, which just return the solution to the problem, as well as expert drivers, which return additional information, such as condition number estimates, and may offer additional facilities such as balancing. The following sub-sections give tables for the driver routines.
3.1.1 Driver routines
3.1.1.1 Linear least squares problems (LLS)
3.1.1.2 Generalized linear least squares problems (LSE and GLM)
3.1.1.3 Symmetric eigenvalue problems (SEP)
3.1.1.4 Nonsymmetric eigenvalue problem (NEP)
3.1.1.5 Singular value decomposition (SVD)
3.1.1.6 Generalized symmetric definite eigenvalue problems (GSEP)
3.1.1.7 Generalized nonsymmetric eigenvalue problem (GNEP)
3.1.1.8 Generalized singular value decomposition (GSVD)
3.1.2 Computational routines
It is possible to solve problems by calling two or more routines in sequence. Some common sequences of routines are indicated in the tables in the following sub-sections; an asterisk (*) against a routine name means that the sequence of calls is illustrated in the example program for that routine.
3.1.2.1 Orthogonal factorizations
Routines are provided for QR factorization (with and without column pivoting), and for LQ, QL and RQ factorizations (without pivoting only), of a general real or complex rectangular matrix. A routine is also provided for the RQ factorization of a real or complex upper trapezoidal matrix. (LAPACK refers to this as the RZ factorization.)
The factorization routines do not form the matrix
Q explicitly, but represent it as a product of elementary reflectors (see
Section 3.3.6). Additional routines are provided to generate all or part of
Q explicitly if it is required, or to apply
Q in its factored form to another matrix (specifically to compute one of the matrix products
QC,
QTC,
CQ or
CQT with
QT replaced by
QH if
C and
Q are complex).
| |
Factorize without pivoting |
Factorize with pivoting |
Generate
Matrix Q |
Apply matrix Q |
| QR factorization,
real matrices |
F08AEF (DGEQRF) |
F08BFF (DGEQP3) |
F08AFF (DORGQR) |
F08AGF (DORMQR) |
| LQ factorization,
real matrices |
F08AHF (DGELQF) |
|
F08AJF (DORGLQ) |
F08AKF (DORMLQ) |
| QL factorization, real matrices |
F08CEF (DGEQLF) |
|
F08CFF (DORGQL) |
F08CGF (DORMQL) |
| RQ factorization, real matrices |
F08CHF (DGERQF) |
|
F08CJF (DORGRQ) |
F08CKF (DORMRQ) |
| RQ factorization, real upper trapezoidal matrices |
F08BHF (DTZRZF) |
|
|
F08BKF (DORMRZ) |
| QR factorization,
complex matrices |
F08ASF (ZGEQRF) |
F08BTF (ZGEQP3) |
F08ATF (ZUNGQR) |
F08AUF (ZUNMQR) |
| LQ factorization,
complex matrices |
F08AVF (ZGELQF) |
|
F08AWF (ZUNGLQ) |
F08AXF (ZUNMLQ) |
| QL factorization,
complex matrices |
F08CSF (ZGEQLF) |
|
F08CTF (ZUNGQL) |
F08CUF (ZUNMQL) |
| RQ factorization,
complex matrices |
F08CVF (ZGERQF) |
|
F08CWF (ZUNGRQ) |
F08CXF (ZUNMRQ) |
| RQ factorization,
complex upper trapezoidal matrices |
F08BVF (ZTZRZF) |
|
|
F08BXF (ZUNMRZ) |
To solve linear least squares problems, as described in
Sections 2.2.1 or
2.2.3, routines based on the
QR factorization can be used:
| real data, full-rank problem |
F08AEF*, F06YJF, F08AGF |
| complex data, full-rank problem |
F08ASF*, F06ZJF, F08AUF |
| real data, rank-deficient problem |
F08BFF*, F06YJF, F08AGF |
| complex data, rank-deficient problem |
F08BTF*, F06ZJF, F08AUF |
To find the minimum norm solution of under-determined systems of linear equations, as described in
Section 2.2.2, routines based on the
LQ factorization can be used:
3.1.2.2 Generalized orthogonal factorizations
Routines are provided for the generalized
QR and
RQ factorizations of real and complex matrix pairs.
3.1.2.3 Singular value problems
Routines are provided to reduce a general real or complex rectangular matrix
A to real bidiagonal form
B by an orthogonal transformation
A=QBPT (or by a unitary transformation
A=QBPH if
A is complex). Different routines allow a full matrix
A to be stored conventionally (see
Section 3.3.1), or a band matrix to use band storage (see
Section 3.3.4 in the F07 Chapter Introduction).
The routines for reducing full matrices do not form the matrix Q or P explicitly; additional routines are provided to generate all or part of them, or to apply them to another matrix, as with the routines for orthogonal factorizations. Explicit generation of Q or P is required before using the bidiagonal QR algorithm to compute left or right singular vectors of A.
The routines for reducing band matrices have options to generate Q or P if required.
Further routines are provided to compute all or part of the singular value decomposition of a real bidiagonal matrix; the same routines can be used to compute the singular value decomposition of a real or complex matrix that has been reduced to bidiagonal form.
Given the singular values,
F08FLF (DDISNA) is provided to compute the reciprocal condition numbers for the left or right singular vectors of a real or complex matrix.
To compute the singular values and vectors of a rectangular matrix, as described in
Section 2.3, use the following sequence of calls:
Rectangular matrix (standard storage)
Rectangular matrix (banded)
To use the singular value decomposition to solve a linear least squares problem, as described in
Section 2.4, the following routines are required:
3.1.2.4 Generalized singular value decomposition
Routines are provided to compute the generalized SVD of a real or complex matrix pair
A,B in upper trapezoidal form. Routines are also provided to reduce a general real or complex matrix pair to the required upper trapezoidal form.
3.1.2.5 Symmetric eigenvalue problems
Routines are provided to reduce a real symmetric or complex Hermitian matrix
A to real tridiagonal form
T by an orthogonal similarity transformation
A=QTQT (or by a unitary transformation
A=QTQH if
A is complex). Different routines allow a full matrix
A to be stored conventionally (see
Section 3.3.1 in the F07 Chapter Introduction) or in packed storage (see
Section 3.3.2 in the F07 Chapter Introduction); or a band matrix to use band storage (see
Section 3.3.4 in the F07 Chapter Introduction).
The routines for reducing full matrices do not form the matrix Q explicitly; additional routines are provided to generate Q, or to apply it to another matrix, as with the routines for orthogonal factorizations. Explicit generation of Q is required before using the QR algorithm to find all the eigenvectors of A; application of Q to another matrix is required after eigenvectors of T have been found by inverse iteration, in order to transform them to eigenvectors of A.
The routines for reducing band matrices have an option to generate Q if required.
Given the eigenvalues,
F08FLF (DDISNA) is provided to compute the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix.
A variety of routines are provided to compute eigenvalues and eigenvectors of the real symmetric tridiagonal matrix T, some computing all eigenvalues and eigenvectors, some computing selected eigenvalues and eigenvectors. The same routines can be used to compute eigenvalues and eigenvectors of a real symmetric or complex Hermitian matrix which has been reduced to tridiagonal form.
Eigenvalues and eigenvectors of real symmetric tridiagonal matrices:
The original (non-reduced) matrix is Real or Complex Hermitian
| all eigenvalues (root-free QR algorithm) |
F08JFF |
| all eigenvalues (root-free QR algorithm called by divide-and-conquer) |
F08JCF or F08JHF |
| all eigenvalues (RRR) |
F08JLF |
| selected eigenvalues (bisection) |
F08JJF |
The original (non-reduced) matrix is Real
| all eigenvalues and eigenvectors (QR algorithm) |
F08JEF |
| all eigenvalues and eigenvectors (divide-and-conquer) |
F08JCF or F08JHF |
| all eigenvalues and eigenvectors (RRR) |
F08JLF |
| all eigenvalues and eigenvectors (positive definite case) |
F08JGF |
| selected eigenvectors (inverse iteration) |
F08JKF |
The original (non-reduced) matrix is Complex Hermitian
| all eigenvalues and eigenvectors (QR algorithm) |
F08JSF |
| all eigenvalues and eigenvectors (divide and conquer) |
F08JVF |
| all eigenvalues and eigenvectors (RRR) |
F08JYF |
| all eigenvalues and eigenvectors (positive definite case) |
F08JUF |
| selected eigenvectors (inverse iteration) |
F08JXF |
The following sequences of calls may be used to compute various combinations of eigenvalues and eigenvectors, as described in
Section 2.7.
Sequences for computing eigenvalues and eigenvectors
Real Symmetric matrix (standard storage)
Real Symmetric matrix (packed storage)
Real Symmetric banded matrix
| all eigenvalues and eigenvectors (using divide-and-conquer) |
F08HCF |
| all eigenvalues and eigenvectors (using QR algorithm) |
F08HEF*, F08JEF |
Complex Hermitian matrix (standard storage)
Complex Hermitian matrix (packed storage)
Complex Hermitian banded matrix
| all eigenvalues and eigenvectors (using divide-and-conquer) |
F08HQF |
| all eigenvalues and eigenvectors (using QR algorithm) |
F08HSF*, F08JSF |
3.1.2.6 Generalized symmetric-definite eigenvalue problems
Routines are provided for reducing each of the problems
Ax=λBx,
ABx=λx or
BAx=λx to an equivalent standard eigenvalue problem
Cy=λy. Different routines allow the matrices to be stored either conventionally or in packed storage. The positive definite matrix
B must first be factorized using a routine from
Chapter F07. There is also a routine which reduces the problem
Ax=λBx where
A and
B are banded, to an equivalent banded standard eigenvalue problem; this uses a split Cholesky factorization for which a routine in
Chapter F08 is provided.
The equivalent standard problem can then be solved using the routines discussed in
Section 3.1.2.5. For example, to compute all the eigenvalues, the following routines must be called:
| real symmetric-definite problem |
F07FDF, F08SEF*, F08FEF, F08JFF |
| real symmetric-definite problem, packed storage |
F07GDF, F08TEF*, F08GEF, F08JFF |
| real symmetric-definite banded problem |
F08UFF*, F08UEF*, F08HEF, F08JFF |
| complex Hermitian-definite problem |
F07FRF, F08SSF*, F08FSF, F08JFF |
| complex Hermitian-definite problem, packed storage |
F07GRF, F08TSF*, F08GSF, F08JFF |
| complex Hermitian-definite banded problem |
F08UTF*, F08USF*, F08HSF, F08JFF |
If eigenvectors are computed, the eigenvectors of the equivalent standard problem must be transformed back to those of the original generalized problem, as indicated in
Section 2.8; routines from
Chapter F06
may be used for this.
3.1.2.7 Nonsymmetric eigenvalue problems
Routines are provided to reduce a general real or complex matrix A to upper Hessenberg form H by an orthogonal similarity transformation A=QHQT (or by a unitary transformation A=QHQH if A is complex).
These routines do not form the matrix Q explicitly; additional routines are provided to generate Q, or to apply it to another matrix, as with the routines for orthogonal factorizations. Explicit generation of Q is required before using the QR algorithm on H to compute the Schur vectors; application of Q to another matrix is needed after eigenvectors of H have been computed by inverse iteration, in order to transform them to eigenvectors of A.
Routines are also provided to balance the matrix before reducing it to Hessenberg form, as described in
Section 2.14.6. Companion routines are required to transform Schur vectors or eigenvectors of the balanced matrix to those of the original matrix.
Routines are provided to compute the eigenvalues and all or part of the Schur factorization of an upper Hessenberg matrix. Eigenvectors may be computed either from the upper Hessenberg form by inverse iteration, or from the Schur form by back-substitution; these approaches are equally satisfactory for computing individual eigenvectors, but the latter may provide a more accurate basis for a subspace spanned by several eigenvectors.
Additional routines estimate the sensitivities of computed eigenvalues and eigenvectors, as discussed in
Section 2.14.5.
Finally routines are provided for reordering the Schur factorization, so that eigenvalues appear in any desired order on the diagonal of the Schur form. The routines
F08QFF (DTREXC) and
F08QTF (ZTREXC) simply swap two diagonal elements or blocks, and may need to be called repeatedly to achieve a desired order. The routines
F08QGF (DTRSEN) and
F08QUF (ZTRSEN) perform the whole reordering process for the important special case where a specified cluster of eigenvalues is to appear at the top of the Schur form; if the Schur vectors are reordered at the same time, they yield an orthonormal basis for the invariant subspace corresponding to the specified cluster of eigenvalues. These routines can also compute the sensitivities of the cluster of eigenvalues and the invariant subspace.
The following sequences of calls may be used to compute various combinations of eigenvalues, Schur vectors and eigenvectors, as described in
Section 2.11:
| real matrix, all eigenvalues and Schur factorization |
F08NEF, F08NFF*, F08PEF |
| real matrix, all eigenvalues and selected eigenvectors |
F08NEF, F08NGF, F08PEF, F08PKF |
| real matrix, all eigenvalues and eigenvectors (with balancing) |
F08NHF*, F08NEF, F08NFF, F08NJF, F08PEF, F08PKF |
| complex matrix, all eigenvalues and Schur factorization |
F08NSF, F08NTF*, F08PSF |
| complex matrix, all eigenvalues and selected eigenvectors |
F08NSF, F08NUF, F08PSF, F08PXF* |
| complex matrix, all eigenvalues and eigenvectors (with balancing) |
F08NVF*, F08NSF, F08NTF, F08NWF, F08PSF, F08PXF |
3.1.2.8 Generalized nonsymmetric eigenvalue problems
Routines are provided to reduce a real or complex matrix pair
A1,R1,
where
A1 is general and
R1 is upper triangular, to generalized upper
Hessenberg form by orthogonal transformations
A1=Q1HZ1T,
R1=Q1RZ1T,
(or by unitary transformations
A1=Q1HZ1H,
R=Q1R1Z1H,
in the complex case). These routines can optionally return
Q1 and/or
Z1. Note that to transform a general matrix pair
A,B to the form
A1,R1 a
QR factorization of
B (
B=Q~R1) should first be performed and the matrix
A1 obtained as
A1=Q~TA (see
Section 3.1.2.1 above).
Routines are also provided to balance a general matrix pair before reducing it to generalized Hessenberg form, as described in
Section 2.14.8. Companion routines are provided to transform vectors of the balanced pair to those of the original matrix pair.
Routines are provided to compute the eigenvalues (as the pairs α,β) and all or part of the generalized Schur factorization of a generalized upper Hessenberg matrix pair. Eigenvectors may be computed from the generalized Schur form by back-substitution.
Additional routines estimate the sensitivities of computed eigenvalues and eigenvectors.
Finally, routines are provided for reordering the generalized Schur factorization so that eigenvalues appear in any desired order on the diagonal of the generalized Schur form.
F08YFF (DTGEXC) and
F08YTF (ZTGEXC) simply swap two diagonal elements or blocks, and may need to be called repeatedly to achieve a desired order.
F08YGF (DTGSEN) and
F08YUF (ZTGSEN) perform the whole reordering process for the important special case where a specified cluster of eigenvalues is to appear at the top of the generalized Schur form; if the Schur vectors are reordered at the same time, they yield an orthonormal basis for the deflating subspace corresponding to the specified cluster of eigenvalues. These routines can also compute the sensitivities of the cluster of eigenvalues and the deflating subspace.
The following sequences of calls may be used to compute various combinations of eigenvalues, generalized Schur vectors and eigenvectors
| real matrix pair, all eigenvalues (with balancing) |
F08AEF, F08AGF, F08WEF, F08WHF, F08XEF* |
| real matrix pair, all eigenvalues and generalized Schur factorization |
F08AEF, F08AFF, F08AGF, F08WEF, F08XEF |
| real matrix pair, all eigenvalues and eigenvectors (with balancing) |
F06QFF, F06QHF, F08AEF, F08AFF, F08AGF, F08WEF, F08WHF, F08XEF, F08YKF*, F08WJF |
| complex matrix pair, all eigenvalues (with balancing) |
F08ASF, F08AUF, F08WSF, F08WVF, F08XSF* |
| complex matrix pair, all eigenvalues and generalized Schur factorization |
F08ASF, F08ATF, F08AUF, F08WSF, F08XSF |
| complex matrix pair, all eigenvalues and eigenvectors (with balancing) |
F06TFF, F06THF, F08ASF, F08ATF, F08AUF, F08WSF, F08WVF, F08XSF, F08YXF*, F08WWF |
3.1.2.9 The Sylvester equation and the generalized Sylvester equation
Routines are provided to solve the real or complex Sylvester equation
AX±XB=C, where
A and
B are upper quasi-triangular if real, or upper triangular if complex. To solve the general form of the Sylvester equation in which
A and
B are general square matrices,
A and
B must be reduced to upper (quasi-) triangular form by the Schur factorization, using routines described in
Section 3.1.2.7. For more details, see the documents for the routines listed below.
Routines are also provided to solve the real or complex generalized Sylvester equations
where the pairs
A,D and
B,E are in generalized Schur form. To solve the general form of the generalized Sylvester equation in which
A,D and
B,E are general matrix pairs,
A,D and
B,E must first be reduced to generalized Schur form.
3.2 NAG Names and LAPACK Names
As well as the NAG routine name (beginning F08), the tables in
Section 3.1 show the LAPACK routine names in double precision.
The routines may be called either by their NAG or 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 F08 routines in the manual normally include the LAPACK double precision names, for example
F08AEF (DGEQRF). The LAPACK routine names follow a simple
scheme (which is similar to that used for the BLAS in
Chapter F06).
Each name has 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) |
|
| – |
the second and third letters YY indicate the type of the matrix A or matrix pair A,B (and in some cases the storage scheme):
| BD |
– bidiagonal |
| DI |
– diagonal |
| GB |
– general band |
| GE |
– general |
| GG |
– general pair (B may be triangular) |
| HG |
– generalized upper Hessenberg |
| HS |
– upper Hessenberg |
| OP |
– (real) orthogonal (packed storage) |
| UP |
– (complex) unitary (packed storage) |
| OR |
– (real) orthogonal |
| UN |
– (complex) unitary |
| PT |
– symmetric or Hermitian positive definite tridiagonal |
| SB |
– (real) symmetric band |
| HB |
– (complex) Hermitian band |
| SP |
– symmetric (packed storage) |
| HP |
– Hermitian (packed storage) |
| ST |
– (real) symmetric tridiagonal |
| SY |
– symmetric |
| HE |
– Hermitian |
| TG |
– triangular pair (one may be quasi-triangular) |
| TR |
– triangular (or quasi-triangular) |
|
| – |
the last three letters ZZZ indicate the computation performed. For example, QRF is a QR factorization. |
Thus the routine
DGEQRF performs a
QR factorization of a real general
matrix; the corresponding routine for a complex general matrix is
ZGEQRF.
3.3 Matrix Storage Schemes
In this chapter the following storage schemes are used for matrices:
- – conventional storage in a two-dimensional array;
- – packed storage for symmetric or Hermitian matrices;
- – packed storage for orthogonal or unitary matrices;
- – band storage for general, symmetric or Hermitian band matrices;
- – storage of bidiagonal, symmetric or Hermitian tridiagonal matrices in two one-dimensional arrays.
These storage schemes are compatible with those used in
Chapters F06 and
F07,
but different schemes for packed, band and tridiagonal storage are used in a few older routines in
Chapters F01,
F02,
F03 and
F04.
3.3.1 Conventional storage
Please see
Section 3.3.1 in the F07 Chapter Introduction for full details.
3.3.2 Packed storage
Please see
Section 3.3.2 in the F07 Chapter Introduction for full details.
3.3.3 Band storage
Please see
Section 3.3.4 in the F07 Chapter Introduction for full details.
3.3.4 Tridiagonal and bidiagonal matrices
A symmetric tridiagonal or bidiagonal matrix is stored in two one-dimensional arrays, one of length
n containing the diagonal elements, and one of length
n-1 containing the off-diagonal elements. (Older routines in
Chapter F02 store the off-diagonal elements in elements
2:n of a vector of length
n.)
3.3.5 Real diagonal elements of complex matrices
Please see
Section 3.3.6 in the F07 Chapter Introduction for full details.
3.3.6 Representation of orthogonal or unitary matrices
A real orthogonal or complex unitary matrix (usually denoted
Q) is often represented in the NAG Library as a product of
elementary reflectors – also referred to as
elementary Householder matrices (usually denoted
Hi). For example,
You need not be aware of the details, because routines are provided to work with this representation, either to generate all or part of
Q explicitly, or to multiply a given matrix by
Q or
QT (
QH in the complex case) without forming
Q explicitly.
Nevertheless, the following further details may occasionally be useful.
An elementary reflector (or elementary Householder matrix)
H of order
n is a unitary matrix of the form
where
τ is a scalar, and
v is an
n element vector, with
τ2v22=2×Reτ;
v is often referred to as the
Householder vector. Often
v has several leading or trailing zero elements, but for the purpose of this discussion assume that
H has no such special structure.
There is some redundancy in the representation
4, which can be removed in various ways. The representation used in
Chapter F08 and in LAPACK (which differs from those used in some of the routines in
Chapters F01,
F02,
F04 and
F06)
sets
v1=1; hence
v1 need not be stored. In real arithmetic,
1≤τ≤2, except that
τ=0 implies
H=I.
In complex arithmetic,
τ may be complex, and satisfies
1≤Reτ≤2 and
τ-1≤1. Thus a complex
H is not Hermitian (as it is in other representations), but it is unitary, which is the important property. The advantage of allowing
τ to be complex is that, given an arbitrary complex vector
x,H can be computed so that
with
real
β. This is useful, for example, when reducing a complex Hermitian matrix to real symmetric tridiagonal form, or a complex rectangular matrix to real bidiagonal form.
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 UPLO='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 SSYTRD ('Upper',...)
3.4.2 Problem dimensions
It is permissible for the problem dimensions (for example, M or N) 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 number of routines implementing block 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 optimal 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 simply 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 optimal block size; this value of LWORK can 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).
If you are in doubt how much workspace to supply and are concerned to achieve optimal 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:
- INFO=0: successful termination;
- 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 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.) In some implementations, especially when linking to vendor versions of LAPACK, execution of the program may continue, in which case, it is essential to test INFO on exit from the routine.
4 Decision Trees
The following decision trees are principally for the computation (general purpose) routines. See
Section 3.1.1.1 for tables of the driver (black box) routines.
4.1 General Purpose Routines (eigenvalues and eigenvectors)
Tree 1: Real Symmetric Eigenvalue Problems
Tree 2: Real Generalized Symmetric-definite Eigenvalue Problems
| Are eigenvalues only required? |
_
yes |
Are all the eigenvalues required? |
_
yes |
Are A and B band matrices? |
_
yes |
F08UFF, F08UEF, F08HEF and F08JFF |
| | |
|
| |
|
no
| |
|
| | |
| | |
|
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GDF, F08TEF, F08GEF and F08JFF |
| | |
|
| |
|
no
| |
|
| | |
| | |
|
F07FDF, F08SEF, F08FEF and F08JFF |
| | |
|
no
| |
|
| | |
|
Are A and B band matrices? |
_
yes |
F08UFF, F08UEF, F08HEF and F08JJF |
| | |
|
no
| |
|
| | |
|
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GDF, F08TEF, F08GEF and F08JJF |
| | |
|
no
| |
|
| | |
|
F07FDF, F08SEF, F08GEF and F08JJF |
no
| |
|
| Are all eigenvalues and eigenvectors required? |
_
yes |
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GDF, F08TEF, F08GEF, F08GFF, F08JEF and F06PLF |
| | |
|
no
| |
|
| | |
|
F07FDF, F08SEF, F08FEF, F08FFF, F08JEF and F06YJF |
no
| |
|
| Are A and B band matrices? |
_
yes |
F08UFF, F08UEF, F08HEF, F08JKF and F06YJF |
no
| |
|
| Are A and B stored with one triangle as a linear array? |
_
yes |
F07GDF, F08TEF, F08GEF, F08JJF, F08JKF, F08GGF and F06PLF |
no
| |
|
| F07FDF, F08SEF, F08FEF, F08JJF, F08JKF, F08FGF and F06YJF |
Note: the routines for band matrices only handle the problem
Ax=λBx; the other routines handle all three types of problems (
Ax=λBx,
ABx=λx or
BAx=λx) except that, if the problem is
BAx=λx and eigenvectors are required,
F06PHF must be used instead of
F06PLF and
F06YFF instead of
F06YJF.
Tree 3: Real Nonsymmetric Eigenvalue Problems
| Are eigenvalues required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PEF |
| | |
|
no
| |
|
| | |
|
F08NAF or F08NBF or (F08NHF, F08NEF and F08PEF) |
no
| |
|
| Is the Schur factorization of A required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PEF |
| | |
|
no
| |
|
| | |
|
F08NBF or (F08NEF, F08NFF, F08PEF or F08NJF) |
no
| |
|
| Are all eigenvectors required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PEF or F08QKF |
| | |
|
no
| |
|
| | |
|
F08NAF or F08NBF or (F08NHF, F08NEF, F08NFF, F08PEF, F08QKF or F08NJF) |
no
| |
|
| Is A an upper Hessenberg matrix? |
_
yes |
F08PEF or F08PKF |
no
| |
|
| F08NHF, F08NEF, F08PEF, F08PKF, F08NGF or F08NJF |
Tree 4: Real Generalized Nonsymmetric Eigenvalue Problems
| Are eigenvalues only required? |
_
yes |
Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XEF |
| | |
|
no
| |
|
| | |
|
F08WHF, F08AEF, F08AGF, F08WEF and F08XEF |
no
| |
|
| Is the generalized Schur factorization of A and B required? |
_
yes |
Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XEF |
| | |
|
no
| |
|
| | |
|
F08AEF, F08AGF, F06QHF, F06QFF, F08AFF, F08WEF, F08XEF and F08YKF |
no
| |
|
| Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XEF and F08YKF |
no
| |
|
| F08WHF, F08AEF, F08AGF, F06QHF, F06QFF, F08AFF, F08WEF, F08XEF, F08YKF and F08WJF |
Tree 5: Complex Hermitian Eigenvalue Problems
Tree 6: Complex Generalized Hermitian-definite Eigenvalue Problems
| Are eigenvalues only required? |
_
yes |
Are all eigenvalues required? |
_
yes |
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GRF, F08TSF, F08GSF and F08JFF |
| | |
|
| |
|
no
| |
|
| | |
| | |
|
F07FRF, F08SSF, F08FSF and F08JFF |
| | |
|
no
| |
|
| | |
|
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GRF, F08TSF, F08GSF and F08JJF |
| | |
|
no
| |
|
| | |
|
F07FRF, F08SSF, F08GSF and F08JJF |
no
| |
|
| Are all eigenvalues and eigenvectors required? |
_
yes |
Are A and B stored with one triangle as a linear array? |
_
yes |
F07GRF, F08TSF, F08GSF, F08GTF and F06PSF |
| | |
|
no
| |
|
| | |
|
F07FRF, F08SSF, F08FSF, F08FTF, F08JSF and F06ZJF |
no
| |
|
| Are A and B stored with one triangle as a linear array? |
_
yes |
F07GRF, F08TSF, F08GSF, F08JJF, F08JXF, F08GUF and F06SLF |
no
| |
|
| F07FRF, F08SSF, F08FSF, F08JJF, F08JXF, F08FUF and F06ZJF |
Tree 7: Complex non-Hermitian Eigenvalue Problems
| Are eigenvalues only required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PSF |
| | |
|
no
| |
|
| | |
|
F08NVF, F08NSF and F08PSF |
no
| |
|
| Is the Schur factorization of A required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PSF |
| | |
|
no
| |
|
| | |
|
F08NSF, F08NTF, F08PSF and F08NWF |
no
| |
|
| Are all eigenvectors required? |
_
yes |
Is A an upper Hessenberg matrix? |
_
yes |
F08PSF and F08QXF |
| | |
|
no
| |
|
| | |
|
F08NVF, F08NSF, F08NTF, F08PSF, F08QXF and F08NWF |
no
| |
|
| Is A an upper Hessenberg matrix? |
_
yes |
F08PSF and F08PXF |
no
| |
|
| F08NVF, F08NSF, F08PSF, F08PXF, F08NUF and F08NWF |
Tree 8: Complex Generalized non-Hermitian Eigenvalue Problems
| Are eigenvalues only required? |
_
yes |
Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XSF |
| | |
|
no
| |
|
| | |
|
F08WVF, F08ASF, F08AUF, F08WSF and F08XSF |
no
| |
|
| Is the generalized Schur factorization of A and B required? |
_
yes |
Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XSF |
| | |
|
no
| |
|
| | |
|
F08ASF, F08AUF, F06THF, F06TFF, F08ATF, F08WSF, F08XSF and F08YXF |
no
| |
|
| Are A and B in generalized upper Hessenberg form? |
_
yes |
F08XSF and F08YXF |
no
| |
|
| F08WVF, F08ASF, F08AUF, F06THF, F06TFF, F08ATF, F08WSF, F08XSF, F08YXF and F08WWF |
4.2 General Purpose Routines (singular value decomposition)
Tree 9
5 Functionality Index
| Backtransformation of eigenvectors from those of balanced forms: | | |
| Backtransformation of generalized eigenvectors from those of balanced forms: | | |
| Eigenvalue problems for condensed forms of matrices: | | |
| complex Hermitian matrix: | | |
| eigenvalues and eigenvectors: | | |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm, using packed storage | | F08HQF (ZHBEVD) |
| all eigenvalues and eigenvectors by root-free QR algorithm | | F08HNF (ZHBEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08HPF (ZHBEVX) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08FQF (ZHEEVD) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm, using packed storage | | F08GQF (ZHPEVD) |
| all eigenvalues and eigenvectors by root-free QR algorithm | | F08FNF (ZHEEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm, using packed storage | | F08GNF (ZHPEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08FPF (ZHEEVX) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration, using packed storage | | F08GPF (ZHPEVX) |
| all eigenvalues and eigenvectors using Relatively Robust Representations or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08FRF (ZHEEVR) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm | | F08HNF (ZHBEV) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection | | F08HPF (ZHBEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, using packed storage | | F08HQF (ZHBEVD) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm | | F08FNF (ZHEEV) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection | | F08FPF (ZHEEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection, using packed storage | | F08GPF (ZHPEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, using packed storage | | F08GNF (ZHPEV) |
| complex upper Hessenberg matrix, reduced from complex general matrix: | | |
| selected right and/or left eigenvectors by inverse iteration | | F08PXF (ZHSEIN) |
| singular value decomposition: | | |
| after reduction from real general matrix, using divide-and-conquer | | F08MDF (DBDSDC) |
| eigenvalues and eigenvectors: | | |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08HCF (DSBEVD) |
| all eigenvalues and eigenvectors by root-free QR algorithm | | F08HAF (DSBEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08HBF (DSBEVX) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08FCF (DSYEVD) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm, using packed storage | | F08GCF (DSPEVD) |
| all eigenvalues and eigenvectors by root-free QR algorithm | | F08FAF (DSYEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm, using packed storage | | F08GAF (DSPEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08FBF (DSYEVX) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration, using packed storage | | F08GBF (DSPEVX) |
| all eigenvalues and eigenvectors using Relatively Robust Representations or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08FDF (DSYEVR) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm | | F08HAF (DSBEV) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection | | F08HBF (DSBEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm | | F08FAF (DSYEV) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection | | F08FBF (DSYEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection, using packed storage | | F08GBF (DSPEVX) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, using packed storage | | F08GAF (DSPEV) |
| real symmetric tridiagonal matrix: | | |
| eigenvalues and eigenvectors: | | |
| after reduction from complex Hermitian matrix: | | |
| all eigenvalues and eigenvectors, using Relatively Robust Representations | | F08JYF (ZSTEGR) |
| all eigenvalues and eigenvectors, using Relatively Robust Representations | | F08JLF (DSTEGR) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08JCF (DSTEVD) |
| all eigenvalues and eigenvectors by root-free QR algorithm | | F08JAF (DSTEV) |
| all eigenvalues and eigenvectors by root-free QR algorithm or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08JBF (DSTEVX) |
| all eigenvalues and eigenvectors using Relatively Robust Representations or selected eigenvalues and eigenvectors by bisection and inverse iteration | | F08JDF (DSTEVR) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm | | F08JAF (DSTEV) |
| all eigenvalues by the Pal–Walker–Kahan variant of the QL or QR algorithm, or selected eigenvalues by bisection | | F08JBF (DSTEVX) |
| real upper Hessenberg matrix, reduced from real general matrix: | | |
| selected right and/or left eigenvectors by inverse iteration | | F08PKF (DHSEIN) |
| Eigenvalue problems for nonsymmetric matrices: | | |
| all eigenvalues, Schur form, Schur vectors and reciprocal condition numbers | | F08PPF (ZGEESX) |
| all eigenvalues and left/right eigenvectors, plus balancing transformation and reciprocal condition numbers | | F08NPF (ZGEEVX) |
| all eigenvalues, real Schur form, Schur vectors and reciprocal condition numbers | | F08PBF (DGEESX) |
| all eigenvalues and left/right eigenvectors, plus balancing transformation and reciprocal condition numbers | | F08NBF (DGEEVX) |
| Eigenvalues and generalized Schur factorization, | | |
| General Gauss–Markov linear model: | | |
| Generalized eigenvalue problems for condensed forms of matrices: | | |
| complex Hermitian-definite eigenproblems: | | |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08UQF (ZHBGVD) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form | | F08UNF (ZHBGV) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form | | F08UPF (ZHBGVX) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08SQF (ZHEGVD) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm, packed storage format | | F08TQF (ZHPGVD) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form | | F08SNF (ZHEGV) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form, packed storage format | | F08TNF (ZHPGV) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form | | F08SPF (ZHEGVX) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form, packed storage format | | F08TPF (ZHPGVX) |
| real symmetric-definite eigenproblems: | | |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08UCF (DSBGVD) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form | | F08UAF (DSBGV) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form | | F08UBF (DSBGVX) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm | | F08SCF (DSYGVD) |
| all eigenvalues and eigenvectors by a divide-and-conquer algorithm, packed storage format | | F08TCF (DSPGVD) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form | | F08SAF (DSYGV) |
| all eigenvalues and eigenvectors by reduction to tridiagonal form, packed storage format | | F08TAF (DSPGV) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form | | F08SBF (DSYGVX) |
| selected eigenvalues and eigenvectors by reduction to tridiagonal form, packed storage format | | F08TBF (DSPGVX) |
| Generalized eigenvalue problems for nonsymmetric matrix pairs: | | |
| complex nonsymmetric matrix pairs: | | |
| all eigenvalues, generalized Schur form, Schur vectors and reciprocal condition numbers | | F08XPF (ZGGESX) |
| all eigenvalues, generalized Schur form and Schur vectors | | F08XNF (ZGGES) |
| all eigenvalues and left/right eigenvectors, plus the balancing transformation and reciprocal condition numbers | | F08WPF (ZGGEVX) |
| real nonsymmetric matrix pairs: | | |
| all eigenvalues, generalized real Schur form and left/right Schur vectors | | F08XAF (DGGES) |
| all eigenvalues, generalized real Schur form and left/right Schur vectors, plus reciprocal condition numbers | | F08XBF (DGGESX) |
| all eigenvalues and left/right eigenvectors, plus the balancing transformation and reciprocal condition numbers | | F08WBF (DGGEVX) |
| Generalized QR factorization: | | |
| Generalized RQ factorization: | | |
| Generalized singular value decomposition | | |
| after reduction from complex general matrix: | | |
| after reduction from real general matrix: | | |
| reduction of a pair of general matrices to triangular or trapezoidal form: | | |
| minimum norm solution using a complete orthogonal factorization | | F08BNF (ZGELSY) |
| minimum norm solution using the singular value decomposition | | F08KNF (ZGELSS) |
| minimum norm solution using the singular value decomposition (divide-and-conquer) | | F08KQF (ZGELSD) |
| reduction of upper trapezoidal matrix to upper triangular form | | F08BVF (ZTZRZF) |
| minimum norm solution using a complete orthogonal factorization | | F08BAF (DGELSY) |
| minimum norm solution using the singular value decomposition | | F08KAF (DGELSS) |
| minimum norm solution using the singular value decomposition (divide-and-conquer) | | F08KCF (DGELSD) |
| reduction of upper trapezoidal matrix to upper triangular form | | F08BHF (DTZRZF) |
| least squares problems with linear equality constraints | | |
| minimum norm solution subject to linear equality constraints using a generalized RQ factorization | | F08ZNF (ZGGLSE) |
| minimum norm solution subject to linear equality constraints using a generalized RQ factorization | | F08ZAF (DGGLSE) |
| Left and right eigenvectors of a pair of matrices: | | |
| LQ factorization and related operations: | | |
| Operations on eigenvectors of a real symmetric or complex Hermitian matrix, or singular vectors of a general matrix: | | |
| Operations on generalized Schur factorization of a general matrix pair: | | |
| estimate condition numbers of eigenvalues and/or eigenvectors | | F08YYF (ZTGSNA) |
| re-order Schur factorization, compute generalized eigenvalues and condition numbers | | F08YUF (ZTGSEN) |
| estimate condition numbers of eigenvalues and/or eigenvectors | | F08YLF (DTGSNA) |
| re-order Schur factorization, compute generalized eigenvalues and condition numbers | | F08YGF (DTGSEN) |
| Operations on Schur factorization of a general matrix: | | |
| re-order Schur factorization, compute basis of invariant subspace, and estimate sensitivities | | F08QUF (ZTRSEN) |
| re-order Schur factorization, compute basis of invariant subspace, and estimate sensitivities | | F08QGF (DTRSEN) |
| Overdetermined and underdetermined linear systems | | |
| solves an overdetermined or undetermined complex linear system | | F08ANF (ZGELS) |
| solves an overdetermined or undetermined real linear system | | F08AAF (DGELS) |
| QL factorization and related operations: | | |
| QR factorization and related operations: | | |
| Reduction of a pair of general matrices to generalized upper Hessenberg form, | | |
| Reduction of eigenvalue problems to condensed forms, and related operations: | | |
| complex general matrix to upper Hessenberg form: | | |
| complex Hermitian band matrix to real symmetric tridiagonal form | | F08HSF (ZHBTRD) |
| complex Hermitian matrix to real symmetric tridiagonal form: | | |
| complex rectangular band matrix to real upper bidiagonal form | | F08LSF (ZGBBRD) |
| complex rectangular matrix to real bidiagonal form: | | |
| real general matrix to upper Hessenberg form: | | |
| real rectangular matrix to bidiagonal form: | | |
| real symmetric matrix to symmetric tridiagonal form: | | |
| Reduction of generalized eigenproblems to standard eigenproblems: | | |
| complex Hermitian-definite banded generalized eigenproblem Ax = λBx | | F08USF (ZHBGST) |
| complex Hermitian-definite generalized eigenproblem Ax = λBx, ABx = λx or BAx = λx | | F08SSF (ZHEGST) |
| complex Hermitian-definite generalized eigenproblem Ax = λBx, ABx = λx or BAx = λx, packed storage | | F08TSF (ZHPGST) |
| real symmetric-definite banded generalized eigenproblem Ax = λBx | | F08UEF (DSBGST) |
| real symmetric-definite generalized eigenproblem Ax = λBx, ABx = λx or BAx = λx | | F08SEF (DSYGST) |
| real symmetric-definite generalized eigenproblem Ax = λBx, ABx = λx or BAx = λx, packed storage | | F08TEF (DSPGST) |
| RQ factorization and related operations: | | |
| Singular value decomposition | | |
| preconditioned Jacobi SVD using fast scaled rotations and de Rijks pivoting | | F08KHF (DGEJSV) |
| Solve generalized Sylvester equation: | | |
| Solve reduced form of Sylvester matrix equation: | | |
| Split Cholesky factorization: | | |
6 Auxiliary Routines Associated with Library Routine Parameters
7 Routines Withdrawn or Scheduled for Withdrawal
None.
8 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
Arioli M, Duff I S and de Rijk P P M (1989) On the augmented system approach to sparse least-squares problems
Numer. Math. 55 667–684
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices
SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems
SIAM J. Numer. Anal. 10 241–256
Parlett B N (1998)
The Symmetric Eigenvalue Problem SIAM, Philadelphia
Stewart G W and Sun J-G (1990)
Matrix Perturbation Theory Academic Press, London
Ward R C (1981) Balancing the generalized eigenvalue problem
SIAM J. Sci. Stat. Comp. 2 141–152
Wilkinson J H (1965)
The Algebraic Eigenvalue Problem Oxford University Press, Oxford
Wilkinson J H and Reinsch C (1971)
Handbook for Automatic Computation II, Linear Algebra Springer–Verlag