LAPACK
LAPACK is a set of Fortran subroutines for a wide range of applications in Linear Algebra. It is
also available in a C translation, CLAPACK, and a Fortran90 version LAPACK90. The LAPACK
users guide is available at
www.netlib.org/lapack/lug/index.html
The following is extracted from the LAPACK users guide.
Problems that LAPACK can Solve
LAPACK can solve systems of linear equations, linear least squares problems, eigenvalue problems
and singular value problems. LAPACK can also handle many associated computations such as
matrix factorizations or estimating condition numbers.
LAPACK contains driver routines for solving standard types of problems, computational
routines to perform a distinct computational task, and auxiliary routines to perform a certain
subtask or common low-level computation. Each driver routine typically calls a sequence of
computational routines. Taken as a whole, the computational routines can perform a wider range of
tasks than are covered by the driver routines. Many of the auxiliary routines may be of use to
numerical analysts or software developers, so we have documented the Fortran source for these
routines with the same level of detail used for the LAPACK routines and driver routines.
Dense and band matrices are provided for, but not general sparse matrices. In all areas, similar
functionality is provided for real and complex matrices. See Chapter 2 for a complete summary of
the contents.
Levels of Routines
The subroutines in LAPACK are classified as follows:

driver routines, each of which solves a complete problem, for example solving a system of
linear equations, or computing the eigenvalues of a real symmetric matrix. Users are
recommended to use a driver routine if there is one that meets their requirements. They are
listed in Section 2.3.

computational routines, each of which performs a distinct computational task, for example
an LU factorization, or the reduction of a real symmetric matrix to tridiagonal form. Each
driver routine calls a sequence of computational routines. Users (especially software
developers) may need to call computational routines directly to perform tasks, or sequences
of tasks, that cannot conveniently be performed by the driver routines. They are listed in
Section 2.4.

auxiliary routines, which in turn can be classified as follows:

routines that perform subtasks of block algorithms -- in particular, routines that
implement unblocked versions of the algorithms;

routines that perform some commonly required low-level computations, for example
scaling a matrix, computing a matrix-norm, or generating an elementary Householder
matrix; some of these may be of interest to numerical analysts or software developers
and could be considered for future additions to the BLAS;

a few extensions to the BLAS, such as routines for applying complex plane rotations,
or matrix-vector operations involving complex symmetric matrices (the BLAS
themselves are not strictly speaking part of LAPACK).
Data Types and Precision
LAPACK provides the same range of functionality for real and complex data.
For most computations there are matching routines, one for real and one for complex data, but there
are a few exceptions. For example, corresponding to the routines for real symmetric indefinite
systems of linear equations, there are routines for complex Hermitian and complex symmetric
systems, because both types of complex systems occur in practical applications. However, there is
no complex analogue of the routine for finding selected eigenvalues of a real symmetric tridiagonal
matrix, because a complex Hermitian matrix can always be reduced to a real symmetric tridiagonal
matrix.
Matching routines for real and complex data have been coded to maintain a close correspondence
between the two, wherever possible. However, in some areas (especially the nonsymmetric
eigenproblem) the correspondence is necessarily weaker.
All routines in LAPACK are provided in both single and double precision versions. The double
precision versions have been generated automatically, using Toolpack/1 [88].
Double precision routines for complex matrices require the non-standard Fortran data type
COMPLEX*16, which is available on most machines where double precision computation is usual.
Naming Scheme
The name of each LAPACK routine is a coded specification of its function (within the very tight
limits of standard Fortran 77 6-character names).
All driver and computational routines have names of the form XYYZZZ, where for some driver
routines the 6th character is blank.
The first letter, X, indicates the data type as follows:
S REAL
D DOUBLE PRECISION
C COMPLEX
Z COMPLEX*16 or DOUBLE COMPLEX
When we wish to refer to an LAPACK routine generically, regardless of data type, we replace the
first letter by ``x''. Thus xGESV refers to any or all of the routines SGESV, CGESV, DGESV and
ZGESV.
The next two letters, YY, indicate the type of matrix (or of the most significant matrix). Most of
these two-letter codes apply to both real and complex matrices; a few apply specifically to one or
the other, as indicated in Table 2.1.
Table 2.1: Matrix types in the LAPACK naming scheme
BD bidiagonal
DI
diagonal
GB general band
GE general (i.e., unsymmetric, in some cases rectangular)
GG general matrices, generalized problem (i.e., a pair of general matrices)
GT general tridiagonal
HB (complex) Hermitian band
HE (complex) Hermitian
HG upper Hessenberg matrix, generalized problem (i.e a Hessenberg and a
triangular matrix)
HP (complex) Hermitian, packed storage
HS upper Hessenberg
OP (real) orthogonal, packed storage
OR (real) orthogonal
PB symmetric or Hermitian positive definite band
PO symmetric or Hermitian positive definite
PP
symmetric or Hermitian positive definite, packed storage
PT symmetric or Hermitian positive definite tridiagonal
SB (real) symmetric band
SP
symmetric, packed storage
ST (real) symmetric tridiagonal
SY symmetric
TB triangular band
TG triangular matrices, generalized problem (i.e., a pair of triangular matrices)
TP triangular, packed storage
TR triangular (or in some cases quasi-triangular)
TZ trapezoidal
UN (complex) unitary
UP (complex) unitary, packed storage
When we wish to refer to a class of routines that performs the same function on different types of
matrices, we replace the first three letters by ``xyy''. Thus xyySVX refers to all the expert driver
routines for systems of linear equations that are listed in Table 2.2.
The last three letters ZZZ indicate the computation performed. Their meanings will be explained in
Section 2.4. For example, SGEBRD is a single precision routine that performs a bidiagonal
reduction (BRD) of a real general matrix.
The names of auxiliary routines follow a similar scheme except that the 2nd and 3rd characters YY
are usually LA (for example, SLASCL or CLARFG). There are two kinds of exception. Auxiliary
routines that implement an unblocked version of a block algorithm have similar names to the
routines that perform the block algorithm, with the sixth character being ``2'' (for example, SGETF2
is the unblocked version of SGETRF). A few routines that may be regarded as extensions to the
BLAS are named according to the BLAS naming schemes (for example, CROT, CSYR).
Driver Routines and Computational Routines





Driver Routines
Linear Equations
Linear Least Squares (LLS) Problems
Generalized Linear Least Squares (LSE and GLM) Problems
Standard Eigenvalue and Singular Value Problems


Symmetric Eigenproblems (SEP)
Nonsymmetric Eigenproblems (NEP)
Singular Value Decomposition (SVD)
Generalized Eigenvalue and Singular Value Problems
 Generalized Symmetric Definite Eigenproblems (GSEP)
Generalized Nonsymmetric Eigenproblems (GNEP)
Generalized Singular Value Decomposition (GSVD)

Computational Routines

Linear Equations
Orthogonal Factorizations and Linear Least Squares Problems
 QR Factorization
LQ Factorization
QR Factorization with Column Pivoting
Complete Orthogonal Factorization
Other Factorizations
Generalized Orthogonal Factorizations and Linear Least Squares Problems
 Generalized QR Factorization
Generalized RQ Factorization
Symmetric Eigenproblems
Nonsymmetric Eigenproblems
 Eigenvalues, Eigenvectors and Schur Factorization
Balancing
Invariant Subspaces and Condition Numbers
Singular Value Decomposition
Generalized Symmetric Definite Eigenproblems
Generalized Nonsymmetric Eigenproblems
 Eigenvalues, Eigenvectors and Generalized Schur Decomposition
Balancing
Deflating Subspaces and Condition Numbers
Generalized (or Quotient) Singular Value Decomposition





















DGETRF(1)
LAPACK routine (version 3.1)
DGETRF(1)
NAME
DGETRF - an LU factorization of a general M-by-N matrix A using partial pivoting with row
interchanges
SYNOPSIS
SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
INTEGER
INFO, LDA, M, N
INTEGER
IPIV( * )
DOUBLE
PRECISION A( LDA, * )
PURPOSE
DGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting
with row interchanges.
The factorization has the form
A=P*L*U
where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower
trapezoidal if m > n), and U is
upper triangular (upper trapezoidal if m < n).
This is the right-looking Level 3 BLAS version of the algorithm.
ARGUMENTS
M
(input) INTEGER
The number of rows of the matrix A. M >= 0.
N
(input) INTEGER
The number of columns of the matrix A. N >= 0.
A
(input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the
factorization A = P*L*U; the
unit diagonal elements of L are not stored.
LDA
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
(output) INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row
IPIV(i).
INFO
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the
factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.
DGETRS(1)
LAPACK routine (version 3.1)
DGETRS(1)
NAME
DGETRS - a system of linear equations A * X = B or A' * X = B with a general N-by-N
matrix A using the LU factorization computed by DGETRF
SYNOPSIS
SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
CHARACTER
TRANS
INTEGER
INFO, LDA, LDB, N, NRHS
INTEGER
IPIV( * )
DOUBLE
PRECISION A( LDA, * ), B( LDB, * )
PURPOSE
DGETRS solves a system of linear equations
A * X = B or A' * X = B with a general N-by-N matrix A using the LU factorization
computed by DGETRF.
ARGUMENTS
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A'* X = B (Transpose)
= 'C': A'* X = B (Conjugate transpose = Transpose)
N
(input) INTEGER
The order of the matrix A. N >= 0.
NRHS
(input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A
(input) DOUBLE PRECISION array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U as computed by DGETRF.
LDA
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
(input) INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the matrix was interchanged with
row IPIV(i).
B
(input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB
(input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
DGEQP3 - compute a QR factorization with column pivoting of a matrix A
SYNOPSIS
SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
INTEGER
INFO, LDA, LWORK, M, N
INTEGER
JPVT( * )
DOUBLE
PRECISION A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
DGEQP3 computes a QR factorization with column pivoting of a matrix A: A*P = Q*R using
Level 3 BLAS.
ARGUMENTS
M
(input) INTEGER
The number of rows of the matrix A. M >= 0.
N
(input) INTEGER
The number of columns of the matrix A. N >= 0.
A
(input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix
R; the elements below the diagonal, together with the array TAU, represent the orthogonal
matrix Q as a product of min(M,N)
elementary reflectors.
LDA
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(J).ne.0, the J-th column of A is permuted to the front of A*P (a
leading column); if JPVT(J)=0, the J-th
column of A is a free column. On exit, if JPVT(J)=K, then the J-th column of A*P was
the the K-th column of A.
TAU
(output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
WORK
(workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO=0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 3*N+1. For optimal performance
LWORK >= 2*N+( N+1 )*NB, where NB is the optimal
blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error message related to LWORK
is issued by XERBLA.
INFO
(output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1;
v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
Based on contributions by
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
X. Sun, Computer Science Dept., Duke University, USA
DORGQR - generate an M-by-N real matrix Q with orthonormal columns,
SYNOPSIS
SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER
INFO, K, LDA, LWORK, M, N
DOUBLE
PRECISION A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
DORGQR generates an M-by-N real matrix Q with orthonormal columns, which is defined as
the first N columns of a product of K elementary reflectors of order M
Q = H(1) H(2) . . . H(k)
as returned by DGEQRF.
ARGUMENTS
M
(input) INTEGER
The number of rows of the matrix Q. M >= 0.
N
(input) INTEGER
The number of columns of the matrix Q. M >= N >= 0.
K
(input) INTEGER
The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.
A
(input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the i-th column must contain the vector which defines the elementary reflector
H(i), for i = 1,2,...,k, as returned
by DGEQRF in the first k columns of its array argument A. On exit, the M-by-N matrix
Q.
LDA
(input) INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU
(input) DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by
DGEQRF.
WORK
(workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N). For optimum performance
LWORK >= N*NB, where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error message related to LWORK
is issued by XERBLA.
INFO
(output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
DGESVD - compute the singular value decomposition (SVD) of a real M-byN matrix A, optionally computing the left and/or right singular vectors
SYNOPSIS
SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
WORK, LWORK, INFO )
CHARACTER
JOBU, JOBVT
INTEGER
INFO, LDA, LDU, LDVT, LWORK, M, N
DOUBLE
PRECISION A( LDA, * ), S( * ), U( LDU, * ), VT(
LDVT, * ), WORK( * )
PURPOSE
DGESVD computes the singular value decomposition (SVD) of a real M-by-N
matrix A, optionally computing the left and/or right singular vectors.
The SVD is written
A = U * SIGMA * transpose(V)
where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N
orthogonal matrix. The diagonal elements of SIGMA are the singular
values of A; they are real and non-negative, and are returned in
descending order. The first min(m,n) columns of U and V are the left
and right singular vectors of A.
Note that the routine returns V**T, not V.
ARGUMENTS
JOBU
(input) CHARACTER*1
Specifies options for computing all or part of the matrix U:
= 'A': all M columns of U are returned in array U:
= 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U; = 'O': the first min(m,n)
columns of U (the left singular vectors) are overwritten on the
array A; = 'N': no columns of U (no left singular vectors) are
computed.
JOBVT (input) CHARACTER*1
Specifies options for computing all or part of the matrix V**T:
= 'A': all N rows of V**T are returned in the array VT;
= 'S': the first min(m,n) rows of V**T (the right singular
vectors) are returned in the array VT; = 'O': the first
min(m,n) rows of V**T (the right singular vectors) are overwritten on the array A; = 'N': no rows of V**T (no right singular vectors) are computed.
JOBVT and JOBU cannot both be 'O'.
M
(input) INTEGER
The number of rows of the input matrix A. M >= 0.
N
(input) INTEGER
The number of columns of the input matrix A. N >= 0.
A
(input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if JOBU = 'O', A is
overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBVT = 'O', A is
overwritten with the first min(m,n) rows of V**T (the right
singular vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT
.ne. 'O', the contents of A are destroyed.
LDA
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
S
(output) DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).
U
(output) DOUBLE PRECISION array, dimension (LDU,UCOL)
(LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU
= 'A', U contains the M-by-M orthogonal matrix U; if JOBU =
'S', U contains the first min(m,n) columns of U (the left sin-
gular vectors, stored columnwise); if JOBU = 'N' or 'O', U is
not referenced.
LDU
(input) INTEGER
The leading dimension of the array U. LDU >= 1; if JOBU = 'S'
or 'A', LDU >= M.
VT
(output) DOUBLE PRECISION array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N orthogonal matrix V**T;
'S', VT contains the first min(m,n) rows of V**T
if JOBVT =
(the right singular vectors, stored rowwise); if JOBVT = 'N' or
'O', VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1; if JOBVT =
'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
WORK
(workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK; if
INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is
in S (not necessarily sorted). B satisfies A = U * B * VT, so
it has the same singular values as A, and singular vectors
related by U and VT.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. LWORK >=
MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,N)).
For good performance,
LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
(output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if DBDSQR did not converge, INFO specifies how many
superdiagonals of an intermediate bidiagonal form B did not
converge to zero. See the description of WORK above for
details.