HELP LAPACK David Young
August 2003
Slightly modified A.Sloman January 2005
LIB * LAPACK provides an interface between Pop-11 and the Lapack and
BLAS libraries of routines for linear algebra (matrix operations).
CONTENTS - (Use g to access required sections)
1 Introduction
2 Installation
2.1 Checking whether Lapack is available from Pop-11
2.2 Installing Lapack
2.3 Telling LIB * LAPACK where to find the libraries
2.4 Testing new installations
3 Scope of the library
4 Documentation
5 Conventions applying to all procedures
5.1 Procedure names
5.2 Argument sequences
5.3 Array-region arguments
5.4 Vector and matrix arguments
5.5 Incremented vector arguments
5.6 Types of array arguments
5.7 The INFO result
5.8 Work arrays
6 Procedures
6.1 BLAS procedures
6.2 Lapack procedures
6.3 Additional procedures
-----------------------------------------------------------------------
1 Introduction
-----------------------------------------------------------------------
Lapack and BLAS together provide a suite of high-quality routines for
linear algebra. For more details, see their web sites:
http://www.netlib.org/lapack/
http://www.netlib.org/blas/
The libraries are freely available, and optimised versions are also
incorporated into some commercial products, such as Sun's Performance
package.
LIB * LAPACK offers a straightforward way to call many of these routines
from Pop-11. It offers:
* loading of the external procedures;
* simplified calling sequences;
* argument consistency checking;
* translation from Pop-11 arrays and array regions to external
representations;
* efficient management of workspace arguments.
In this file, the external Fortran library is referred to as Lapack, and
the Pop-11 interface library as LAPACK.
A higher-level interface is provided by *LAPOP.
-----------------------------------------------------------------------
2 Installation
-----------------------------------------------------------------------
2.1 Checking whether Lapack is available from Pop-11
-----------------------------------------------------
To see whether Lapack is available, try the following
uses popvision, lapack;
vars x = newsfloatarray([1 5], identfn);
xDOT(x,[],false, x,[],false) =>
If there are no warnings or mishaps and "** 55.0" is printed, Lapack is
available on your system and you can skip the rest of this section.
If there were warnings or mishaps, it may be necessary to install Lapack
and BLAS, to tell LIB * LAPACK where to find them, or both.
2.2 Installing Lapack
----------------------
Lapack may be installed in different places on different systems. You
should check with the documentation or a system manager to find out
where the libraries should be on your system - they may be already
installed.
If they are not, you can download Lapack and BLAS from the sites
mentioned above, and carry out an installation using the documentation
provided.
2.3 Telling LIB * LAPACK where to find the libraries
-----------------------------------------------------
If Lapack and BLAS have been installed, it is worth running the test
above again to see if the library can now find them. If there is still a
mishap, you can specify their location by setting the variable
lapack_libraries to a list of strings, giving the linker options
necessary to locate the libraries and associated routines such as the
Fortran run-time library. This forms the input file list to exload - see
REF * EXTERNAL for further guidance.
For example on a Sun Solaris system with the Performance package
installed, the correct setting is
vars lapack_libraries =
['-L/opt/SUNWspro/lib' '-lsunperf' '-lF77' '-lM77' '-lsunmath'];
- though this case is already built in to the system.
This is the linux version, added at Birmingham 7 Jan 2005
You may need something different.
vars lapack_libraries =
;;; link options obtained from running f77 -v
['-L/usr/lib'
'-L/usr/lib/gcc-lib/i386-redhat-linux/3.2.2/'
'-llapack'
;;; Not sure if this is needed
;;; '-lg2c'
'-lm'
;;; Not sure if this is needed
;;; '-lgcc_s'
];
It may be difficult to get this list correct. Problems may either be
manifested by compile-time errors when libraries are not found, or with
dynamic linking by run-time errors, when particular symbols (usually
procedure names) cannot be resolved. It may be helpful to see what
libraries are linked when a Fortran program that calls a Lapack routine
is compiled, linked and run - there may be flags to the compiler-driver
that allow you to see this.
The variable must be set before LAPACK is loaded.
Alternatively, the LAPACK library file itself may be edited to include
the correct file list for your system. The conditional statements under
"External Routines" should be changed to add a test that identifies your
system and sets up the variable as required. You could also let the
author (d.s.young@sussex.ac.uk) know so that master copies of the
library can be updated.
Please also inform A.Sloman@cs.bham.ac.uk so that the version at the
Poplog distribution site can be updated.
2.4 Testing new installations
------------------------------
If the library is linked to a new installation of Lapack, it is
important to run test examples and to check that the results returned
are as expected, to ensure that the conventions for Fortran argument
passing are being respected.
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3 Scope of the library
-----------------------------------------------------------------------
The library does not provide an interface to every Lapack or BLAS
routine. At the time of writing, access to 164 external procedures is
given through 61 Pop-11 procedures. Single-precision real,
double-precision real, single-precision complex and double-precision
complex cases are included.
Routines omitted are, by and large, those appearing in Lapack 3.0 or
later and those that use specialised storage schemes such as packed
symmetric matrices with variable length columns.
The library can readily be extended to include more recent routines.
If it is extended to cover special storage cases, it would be good also
to provide Pop-11 support for accessing these.
A few procedures are provided which do not simply call a single
Lapack/BLAS routine. These provide matrix functions which cannot be
carried out so efficiently at a higher level, because of the calling
conventions for LAPACK. They are generally cases where a Lapack vector
routine needs to be called in a loop to operate on a whole matrix.
-----------------------------------------------------------------------
4 Documentation
-----------------------------------------------------------------------
The individual routines are not documented within Poplog (apart from the
few which do not map directly onto Lapack routines, described below).
Probably the best sources of documentation are the web sites mentioned
above. The Lapack Users' Guide is also available in print.
Documentation may also be installed with the routines. On Unix systems,
MAN pages should be available. To access these, replace the "x" in the
Pop-11 routine name with "S" (or in cases where the routine only takes
complex arguments, replace it with "C"). For example, to learn what xDOT
does, the ved command "man SDOT" may usually be used; for xDOTC
do "man CDOTC".
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5 Conventions applying to all procedures
-----------------------------------------------------------------------
5.1 Procedure names
--------------------
Pop-11 procedure names are derived from the external procedure names by
replacing the "precision letter" - S, D, C or Z - with "x". Usually this
is the initial character of the name (though IxAMAX is an exception).
The actual external procedure called will depend on the type of the
array arguments.
5.2 Argument sequences
-----------------------
The argument sequences are as for the external procedures, except that:
* CHARACTER*1 options are replaced by Pop-11 strings;
* each array argument is followed by a region specification - see
below;
* problem size arguments (N, NRHS etc.) are omitted;
* leading dimension arguments (LDA etc.) are omitted;
* workspace arguments are omitted;
* INFO is omitted;
* scalar arguments that are output-only are omitted;
* function results, and the values of output scalar arguments, are
returned on the stack.
In general each Pop-11 routine has a single argument sequence. However
there are certain routines where the sequence changes, depending on the
argument types - for example xGEEV. This is because the external
procedures require different sequences for real and complex data, and it
was felt better to reflect this in the interface rather than break the
rules above.
5.3 Array-region arguments
---------------------------
Wherever an array appears as an argument to an external routine, the
Pop-11 procedure requires two arguments, an array and a region
specification. The region spec is a list giving the region of the array
to process. It has the form
[element1 element2]
or
[row1 row2 col1 col2]
for 1-D and 2-D arrays respectively, giving the minimum and maximum
subscripts in each dimension. This is the same as the form of the
*boundslist used to specify array dimensions when an array is created.
The region spec may be the empty list []. This is equivalent to giving
the boundslist of the array (i.e. the whole array is used), but is
handled more efficiently.
For efficiency, array data are never copied - a pointer to the array, or
an element of it, is passed to the external routine, even if only part
of the array is being processed. This means that if an external
procedure changes the data in an array, it will be changed on exit from
the Pop-11 procedure. Thus you cannot necessarily expect arrays that you
might think of as inputs to be unchanged after a call to a LAPACK
routine - you need to check the documentation.
For this reason, no distinction is made in the interface between input
and output arrays. You have to supply all the arrays required, and
whether the elements are accessed, updated or both is determined by the
external routine.
5.4 Vector and matrix arguments
--------------------------------
Lapack/BLAS routines expect either 1-D vectors or 2-D matrices. There is
some flexibility allowed in how these are presented:
* A 1-D array may be supplied where a 2-D array is expected, and is
treated as a 2-D array with one column.
* A 2-D array may be supplied where a 1-D array is expected,
provided the region specified contains only a single column, or part
of one.
Note that the first subscript refers to the row, and the second to
the column.
5.5 Incremented vector arguments
---------------------------------
Some BLAS routines expect 1-D vectors to be followed by an integer
increment, which specifies that only every N'th element of the vector
should be accessed. When calling the LAPACK procedure, the equivalent
argument, which must be an integer or , is given after the region
spec. There are various possibilities:
* If the increment is an integer N, the corresponding array/region
argument should be as for a 1-D input. Every N'th element, starting
with the first, will be processed.
* If the increment is , and the array/region is a row (or
part of a row) of a 2-D array, that row (or part) will be processed.
(The external routine is actually passed the leading dimension of
the array as the increment.)
* If the increment is and the array/region contains a single
column, or part of one, an increment of 1 is assumed.
* Other cases will cause a mishap.
5.6 Types of array arguments
-----------------------------
Floating point array arguments must be "packed" arrays (that is, made
specially to contain numerical data, not ordinary Pop-11 arrays which
have a "full" arrayvector). They must be created with the variable
*poparray_by_row set to (that is, they are stored "by column",
oddly enough).
You can create suitable arrays using the procedures in *popvision:
*newsfloatarray - single precision real
*newdfloatarray - double precision real
*newcfloatarray - single precision complex
*newzfloatarray - double precision complex
You can also generate suitable arrays using *defclass and *newanyarray.
Arrays that are to be treated as complex must have either "cfloatvec" or
"zfloatvec" as the dataword of the arrayvector, depending on the
precision - see HELP * EXCALL for more information.
All the floating point array arguments in a call to a LAPACK procedure
must be the same type, except where the specification calls for a mix of
real and complex arguments. In the latter case, each argument must be
real or complex as required, and they must all have the same precision.
Integer array arguments must be packed arrays of integers. They can be
created using the popvision procedure *newintarray.
5.7 The INFO result
--------------------
Lapack (but not BLAS) routines return an INFO result. If this is
negative, it indicates an error in an argument, and this will cause a
mishap. If it is positive, it indicates a numerical problem. This will
trigger an exception, which may be trapped but otherwise will just print
a warning.
The procedure lapack_lasterror will return the most recent INFO result:
lapack_lasterror() -> int
5.8 Work arrays
----------------
Arrays whose primary purpose is workspace are omitted from the Pop-11
routines' argument sequences.
Work arrays are created using *oldarray and are recycled. Each procedure
maintains its own work arrays, which are reused if the procedure is
called again for the same size of problem before the next garbage
collection. If the procedure is called for a smaller problem, the
arrayvectors are reused.
Where necessary, ILAENV is called to find the optimum work array
dimensions. Where the documentation does not specify an algorithm to
determine the optimum size, the minimum size is used on the first call
to a given routine with given problem dimensions. On return, the first
element of the work array is inspected to determine the optimum size,
and this is used on subsequent calls.
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6 Procedures
-----------------------------------------------------------------------
6.1 BLAS procedures
--------------------
xSWAP(x, xr, incx, y, yr, incy)
Exchange vectors
xSCAL(alpha, y, yr, incy)
Scale vector
xCOPY(x, xr, incx, y, yr, incy)
Copy vector
xAXPY(alpha, x, xr, incx, y, yr, incy)
alpha * x + y
xDOT(x, xr, incx, y, yr, incy) -> result
Real dot product
xDOTU(x, xr, incx, y, yr, incy) -> result
Complex dot product
xDOTC(x, xr, incx, y, yr, incy) -> result
Dot product of x and conjg(y)
xNRM2(x, xr, incx) -> result
Euclidean norm
xASUM(x, xr, incx) -> result
Sum of absolute values
IxAMAX(x, xr, incx) -> result
Index of element with largest absolute value (abs(realpart) +
abs(imagpart))
xGEMV(trans, alpha, a, ar, x, xr, incx, beta, y, yr, incy)
Matrix-vector product alpha*A*x + beta*y or alpha*A'*x + beta*y
xGER(alpha, x, xr, incx, y, yr, incy, a, ar)
Outer product alpha * x * y' * A (real)
xGERU(alpha, x, xr, incx, y, yr, incy, a, ar)
Outer product alpha * x * y' * A (complex)
xGERC(alpha, x, xr, incx, y, yr, incy, a, ar)
Outer product alpha * x * conjg(y') * A (complex)
xSYR(uplo, alpha, x, xr, incx, a, ar)
alpha * x * x' + A (real)
xHER(uplo, alpha, x, xr, incx, a, ar)
alpha * x * conjg(x') + A (complex)
xSYR2(uplo, alpha, x, xr, incx, y, yr, incy, a, ar)
alpha * (x*y' + y*x') + A (real)
xHER2(uplo, alpha, x, xr, incx, y, yr, incy, a, ar)
alpha*x*conjg(y') + conjg(alpha)*y*conjg(x') + A (complex)
xGEMM(transa, transb, alpha, a, ar, b, br, beta, c, cr)
Matrix-matrix product alpha*A[']*B['] + beta*C
xSYMM(side, uplo, alpha, a, ar, b, br, beta, c, cr)
alpha*A*B + beta*C or alpha*B*A + beta*C, A symmetric
xHEMM(side, uplo, alpha, a, ar, b, br, beta, c, cr)
alpha*A*B + beta*C or alpha*B*A + beta*C, A hermitian
xSYRK(uplo, trans, alpha, a, ar, beta, c, cr)
alpha*A*A' + beta*C or alpha*A'*A + beta*C
xHERK(uplo, trans, alpha, a, ar, beta, c, cr)
alpha*A*conjg(A') + beta*C or alpha*conjg(A')*A + beta*C
(complex)
xSYR2K(uplo, trans, alpha, a, ar, b, br, beta, c, cr)
alpha*A*B'+alpha*B*A'+beta*C or alpha*A'*B+alpha*B'*A+beta*C
xHER2K(uplo, trans, alpha, a, ar, b, br, beta, c, cr)
alpha*A*conjg(B') + alpha*B*conjg(A') + beta*C
or alpha*conjg(A')*B+alpha*conjg(B')*A+beta*C (complex)
6.2 Lapack procedures
----------------------
xLACGV(x, xr, incx)
Complex conjugate vector (complex)
xLACPY(uplo, a,ar, b,br)
Copy matrix
xGESV(a,ar, ipiv,ipivr, b,br)
Solve A*X = B, A square
xSYSV(uplo, a,ar, ipiv,ipivr, b,br)
Solve A*X = B, A symmetric
xHESV(uplo, a,ar, ipiv,ipivr, b,br)
Solve A*X = B, A hermitian (complex)
xPOSV(uplo, a,ar, b,br)
Solve A*X = B, A symmetric positive definite
xGESVX(fact, trans, a,ar, af,afr, ipiv,ipivr, equed, r,rr,
c,cr, b,br, x,xr, ferr,ferrr, berr,berrr) -> rcond
Solve A*X = B, expert driver
xSYSVX(fact, uplo, a,ar, af,afr, ipiv,ipivr,
b,br, x,xr, ferr,ferrr, berr,berrr) -> rcond
Solve A*X = B, A symmetric, expert driver
xHESVX(fact, uplo, a,ar, af,afr, ipiv,ipivr,
b,br, x,xr, ferr,ferrr, berr,berrr) -> rcond
Solve A*X = B, A hermitian, expert driver (complex)
xPOSVX(fact, uplo, a,ar, af,afr, equed, s,sr,
b,br, x,xr, ferr,ferrr, berr,berrr) -> rcond
Solve A*X = B, A symmetric positive definite, expert driver
xGELS(trans, a,ar, b,br)
Solve A*X = B, least squares solution
xGELSX(a,ar, b,br, jpvt,jpvtr, rcond) -> rank
Solve A*X = B, least squares solution, expert driver
xGELSS(a,ar, b,br, s,sr, rcond) -> rank
Solve A*X = B, least squares solution, using SVD
xGGLSE(a,ar, b,br, c,cr, d,dr, x,xr)
Minimise ||c - A*x|| subject to B*x = d
xGGGLM(a,ar, b,br, d,dr, x,xr, y,yr)
Solve Gauss-Markov linear model: find x to minimise ||y||
subject to A*x + B*y = d
xSYEV(jobz, uplo, a, ar, w, wr)
Eigenvalues and eigenvectors of symmetric matrix (real)
xHEEV(jobz, uplo, a, ar, w, wr)
Eigenvalues and eigenvectors of hermitian matrix (complex)
xSYEVD(jobz, uplo, a, ar, w, wr)
Eigenvalues and eigenvectors of symmetric matrix using
divide-and-conquer (real)
xHEEVD(jobz, uplo, a, ar, w, wr)
Eigenvalues and eigenvectors of hermitian matrix using
divide-and-conquer (complex)
xSYEVX(jobz, range, uplo, a,ar, vl,vu, il,iu,
abstol, w,wr, z,zr, ifail,ifailr, func) -> m
Eigenvalues and eigenvectors of symmetric matrix, expert driver
(real)
xHEEVX(jobz, range, uplo, a,ar, vl,vu, il,iu,
abstol, w,wr, z,zr, ifail,ifailr, func) -> m
Eigenvalues and eigenvectors of hermitian matrix, expert driver
(complex)
xGEEV(jobvl, jobvr, a,ar, w,wr, [wi, wir], vl,vlr, vr,vrr)
Eigenvalues and eigenvectors of square nonsymmetric matrix
xGEEVX(balanc, jobvl, jobvr, sense, a,ar, w,wr, [wi, wir],
vl,vlr, vr,vrr, scale,scaler, rconde,rconder, rcondv,rcondvr)
-> (ilo, ihi, abnrm)
Eigenvalues and eigenvectors of square nonsymmetric matrix,
expert driver
xGESVD(jobu, jobvt, a, ar, s, sr, u, ur, vt, vtr)
Singular value decomposition
xSYGV(itype, jobz, uplo, a,ar, b,br, w,wr)
Generalised eigenproblem: A*x = lambda*B*x or A*B*x = lambda*x
or B*A*x = lambda*x, A symmetric, B symmetric positive definite
(real)
xHEGV(itype, jobz, uplo, a,ar, b,br, w,wr)
Generalised eigenproblem: A*x = lambda*B*x or A*B*x = lambda*x
or B*A*x = lambda*x, A hermitian, B hermitian positive definite
(complex)
xGEGV(jobvl, jobvr, a,ar, b,br, alpha,alphar, [alphai, alphair],
beta,betar, vl,vlr, vr,vrr)
Generalised eigenvalues and eigenvectors: (A - wB)*r = 0 where w
= alpha/beta, or conjg(l')*(A - wB) = 0
xGGSVD(jobu, jobv, jobq, a,ar, b,br, alpha,alphar, beta,betar,
u,ur, v,vr, q,qr) -> (k, l)
Generalised singular value decomposition: U'*A*Q = D1*(0 R),
V'*B*Q = D2*(0 R)
6.3 Additional procedures
--------------------------
xLPCOL(a,ar,ia, b,br,ib)
Copies column IA of A to column IB of B. IA and IB are matrix
indices; that is the leftmost column of each array region is
column 1, regardless of the underlying array indices.
xLPROW(a,ar,ja, b,br,jb)
Copies row JA of A to row JB of B JA and JB are matrix indices;
that is the top row of each array region is row 1, regardless of
the underlying array indices.
xLPTRANS(a,ar, b,br)
Transposes a matrix
xLPADJOINT(a,ar, b,br)
Forms the adjoint (the complex conjugate of the transpose) of a
matrix (complex)
xLPRESHAPE(aorder, border, a,ar, b,br)
Copies sequentially from A to B, taking elements from A by
column if the string AORDER starts with 'c' or 'C' and by row if
it starts with 'r' or 'R', and writing into B with the order
similarly determined by BORDER
xLPRITOC(a,ar, b,br, c,cr)
Copies real matrix A to the real part of complex matrix C, and
real matrix B to the imaginary part of C
xLPCTORI(c,cr a,ar, b,br)
Copies the real part of complex matrix C to the real matrix A,
and the imaginary part of C to the real matrix B
xLPRTOC(a,ar, c,cr)
Copies real matrix A to the real part of complex matrix C, and
sets the imaginary part of C to zero
xLPSCAL(alpha, a,ar)
Multiplies each element of A by the scalar alpha
xLPAXPY(alpha, a,ar, b,br)
Assigns alpha*A+B to B
--- $popvision/help/lapack
--- Copyright University of Sussex 2005. All rights reserved.