CXML
dsskyf
Symmetric sparse matrix factorization using skyline storage scheme
(Serial and Parallel Versions)
FORMAT
DSSKYF
(n, au, iaudiag, nau, iparam, rparam, iwrk, rwrk, ierror)
Arguments
n integer*4
On entry, the order of the matrix A.
On exit, n is unchanged.
au real*8
On entry, an array of length at least nau, containing
the matrix A stored in the skyline storage scheme,
using either the profile-in or the diagonal-out storage
mode.
On exit, au contains the transp(U)*D*U factorization
of the matrix A. au must remain unchanged between the
call to the routine DSSKYF and any routine that uses
the factors such as DSSKYS, DSSKYC, DSSKYR and DSSKYX.
iaudiag integer*4
On entry, an array of length at least n for the
profile-in storage mode and n+1 for the diagonal-out
storage mode, containing the pointers to the locations
of the diagonal elements in array AU.
On exit, iaudiag is unchanged.
nau integer*4
On entry, the number of elements in array AU. nau is
also the envelope size of the symmetric part of the
matrix A. For the profile-in storage mode, nau =
IAUDIAG(n). For the diagonal-out storage mode, nau =
IAUDIAG(n+1) - 1.
On exit, nau is unchanged.
iparam integer*4
An array of length at least 100, containing the integer
parameters for the transp(U)*D*U factorization.
iparam(1): niparam
On entry, defines the length of the array IPARAM.
niparam >= 100.
On exit, iparam(1) is unchanged.
iparam(2): nrparam
On entry, defines the length of the array RPARAM.
nrparam >= 100.
On exit, iparam(2) is unchanged.
iparam(3): niwrk
On entry, defines the size of the integer work array,
IWRK. niwrk >=2n.
On exit, iparam(3) is unchanged.
iparam(4): nrwrk
On entry, defines the size of the real work array,
RWRK. As the real work array is not used at present,
nrwrk can be unspecified.
On exit, iparam(4) is unchanged.
iparam(5): iounit
On entry, defines the I/O unit number for printing
error messages and information from the routine DSSKYF.
The I/O unit must be opened in the calling subprogram.
If iounit <= 0, no output is generated.
On exit, iparam(5) is unchanged.
iparam(6): iolevel
On entry, defines the message level that determines the
amount of information printed out to iounit, when
iounit > 0.
iolevel = 0 : fatal error messages only
iolevel = 1 : error messages and minimal information
iolevel = 2 : error messages and detailed information
On exit, iparam(6) is unchanged.
iparam(7): idefault
On entry, defines if the default values should be used
in arrays IPARAM and RPARAM. If idefault = 0, then the
following default values are assigned:
IPARAM(1) = niparam = 100
IPARAM(2) = nrparam = 100
IPARAM(6) = iolevel = 0
IPARAM(8) = istore = 1
IPARAM(9) = ibeg = 0
IPARAM(10) = idet = 0
IPARAM(11) = ipvt = 0
IPARAM(13) = inertia = 0
RPARAM(1) = pvt_sml = 10**(-12)
If idefault = 1, then you must assign values to the
above variables before the call to the DSSKYF routine.
On exit, iparam(7) is unchanged.
iparam(8): istore
On entry, defines the type of storage scheme used for
the skyline matrix. If istore = 1, the matrix A is
stored using the profile-in storage mode; if istore =
2, the matrix A is stored using the diagonal-out
storage mode. Default: istore = 1.
On exit, iparam(8) is unchanged.
iparam(9): ibeg
On entry, defines if full or partial factorization is
to be performed. If ibeg = 0, then a full
factorization is performed for rows and columns 1
through n. If ibeg > 0, then a partial factorization
is performed starting from rows and columns ibeg+1
through n, that is, rows and columns from 1 through
ibeg have already been factorized. Default: ibeg = 0.
On exit, iparam(9) is unchanged.
iparam(10): idet
On entry, defines if the determinant of the matrix A is
to be calculated. If idet = 0, then the determinant
is not calculated; if idet = 1, the determinant is
calculated as det_base * 10**det_pwr See RPARAM(4) and
RPARAM(5). Default: idet = 0.
On exit, iparam(10) is unchanged.
iparam(11): ipvt
On entry, defines if the factorization should continue
when a small pivot (defined by RPARAM(1)) is
encountered. If ipvt = 0 and the absolute value of the
pivot element is smaller than pvt_sml = RPARAM(1),
then the factorization process is stopped and control
returned to the calling sub-program. If ipvt = 1 and
a pivot smaller than RPARAM(1) in absolute value is
encountered in the factorization, the process
continues. If ipvt = 2 and a pivot smaller than
RPARAM(1) in absolute value is encountered in the
factorization, it is replaced by a predetermined value
pvt_new = RPARAM(2), and the factorization is
continued. Default: ipvt = 0.
On exit, iparam(11) is unchanged.
iparam(12): ipvt_loc
On entry, an unspecified variable.
On exit, iparam(12) contains the location of the first
pivot element smaller in absolute value than pvt_sml.
The pivot element is returned in pvt_val = RPARAM(3).
If iparam(12) = 0, then no such pivot element exists.
iparam(13): inertia
On entry, defines if the inertia of the matrix A should
be calculated. The inertia of the symmetric matrix A is
the triplet of integers (ipeigen, ineigen, izeigen),
consisting of the number of positive, negative and zero
eigenvalues, respectively. If inertia = 0, then the
inertia is not calculated; if inertia = 1, then the
number of positive and negative eigenvalues are
returned in ipeigen = IPARAM(14) and ineigen =
IPARAM(15), respectively. An indication of the
existence of zero eigenvalues is returned in izeigen =
IPARAM(16). Default: inertia = 0.
On exit, iparam(13) is unchanged.
iparam(14): ipeigen
On entry, an unspecified variable.
On exit, if inertia = 1, iparam(14) contains the
number of positive eigenvalues of the matrix A.
iparam(15): ineigen
On entry, an unspecified variable.
On exit, if inertia = 1, iparam(15) contains the
number of negative eigenvalues of the matrix A.
iparam(16): izeigen
On entry, an unspecified variable.
On exit, if inertia = 1, iparam(16) indicates if the
matrix A has any zero eigenvalues. If izeigen = 0,
then the matrix A does not have a zero eigenvalue; if
izeigen = 1, then the matrix A has at least one zero
eigenvalue.
rparam real*8
An array of length at least 100, containing the real
parameters for the transp(U)*D*U factorization.
rparam(1): pvt_sml
On entry, defines the value of the pivot element which
is considered to be small. If a pivot element smaller
than pvt_sml, in absolute value, is encountered in the
factorization process, then, depending on the value of
ipvt = IPARAM(11), the process either stops, continues
or continues after the pivot is set equal to pvt_new =
RPARAM(2). pvt_sml > 0. Recommended value: 10**(-15)
<= pvt_sml <= 1. Default: pvt_sml = 10**(-12).
On exit, rparam(1) is unchanged.
rparam(2): pvt_new
On entry, defines the value to which the pivot element
must be set if ipvt = 2 and the pivot element is less
than pvt_sml in absolute value. pvt_new should be
large enough to avoid overflow when calculating the
reciprocal of the pivot element.
On exit, rparam(2) is unchanged.
rparam(3): pvt_val
On entry, an unspecified variable.
On exit, rparam(3) contains the value of the first
pivot element smaller than pvt_sml in absolute value.
This element occurs at the location returned in
IPARAM(12). If no such pivot element is found, the
value of pvt_val is unspecified.
rparam(4): det_base
On entry, an unspecified variable.
On exit, defines the base for the determinant of the
matrix A. If idet = 1, the determinant is calculated
as det_base * 10**(det_pwr). 1.0 <= det_base < 10.0.
rparam(5): det_pwr
On entry, an unspecified variable.
On exit, defines the power for the determinant of the
matrix A. If idet = 1, the determinant is calculated
as det_base * 10**det_pwr.
iwrk integer*4
On entry, an array of length at least 2n used for
integer workspace.
On exit, iwrk contains information for use by routines
that use the factorization such as DSSKYS, DSSKYC,
DSSKYR and DSSKYX. The first 2n elements of iwrk
should therefore be passed unchanged to these routines.
rwrk real*8
On entry, an array used for real workspace.
On exit, rwrk is unchanged. Presently, rwrk is not
used by the routine DSSKYF. It can be a dummy
variable.
ierror integer*4
On entry, an unspecified variable.
On exit, ierror contains the error flag. A value of
zero indicates a normal exit from the routine DSSKYF.
Description
DSSKYF obtains the factorization of the symmetric matrix A as:
A = transp(U)*D*U
where D is a diagonal matrix, and U is a unit upper triangular matrix. The
matrix A is stored in a skyline form, using either the profile-in storage
mode or the diagonal-out storage mode.
The routine DSSKYF does not perform any pivoting to preserve the numerical
stability of the transp(U)*D*U factorization. It is therefore primarily
intended for the solution of symmetric positive (or negative) definite
systems as they do not require pivoting for numerical stability. Caution is
urged when using this routine for symmetric indefinite systems.
If a small pivot, in absolute value, pvt_sml, is encountered in the
process of factorization, you have the option of either stopping the
factorization process and returning to the calling sub-program, continuing
the factorization process with the small value of the pivot, or continuing
after setting the pivot equal to some predetermined value, pvt_new. The
location of the first occurrence of a small pivot is returned in ipvt_loc
and its value in pvt_val.
In addition to the transp(U)*D*U factorization, you can also obtain the
determinant of A, the number of positive and negative eigenvalues of the
matrix A and an indication of the existence of zero eigenvalues. A partial
factorization can also be obtained by appropriately setting the value of
ibeg. If ibeg > 0, then factorization begins at row and column ibeg + 1;
the rows and columns from 1 to ibeg are assumed to have been already
factorized. When ibeg > 0, the determinant of A, the inertia of A, and the
statistics on the matrix are calculated from rows and columns ibeg + 1
through n. If the factorization process is stopped at row i due to a small
pivot, then the inertia, determinant and statistics on the matrix are
evaluated for rows ibeg + 1 through (i-1).
The data in the first 2n elements of the integer workspace, IWRK, are used
in routines that use the transp(U)*D*U factorization, such as DSSKYS,
DSSKYC and DSSKYR. This data must therefore remain unchanged between the
call to DSSKYF and any one of these routines. The real workspace array,
RWRK, is not used at present.
This routine is available in both serial and parallel versions. The routine
names and parameter list are identical for both versions. For information
about linking to the serial or to the parallel library, refer to the CXML
Reference Manual.
CXML Home Page Index of CXML Routines