dgbsvx man page on Scientific

```DGBSVX(1)	      LAPACK driver routine (version 3.2)	     DGBSVX(1)

NAME
DGBSVX  -  uses	the LU factorization to compute the solution to a real
system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

SYNOPSIS
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,	LDAFB,
IPIV,	 EQUED,	 R,  C,	 B,  LDB, X, LDX, RCOND, FERR,
BERR, WORK, IWORK, INFO )

CHARACTER	  EQUED, FACT, TRANS

INTEGER	  INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

DOUBLE	  PRECISION RCOND

INTEGER	  IPIV( * ), IWORK( * )

DOUBLE	  PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,  *
),  BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ),
X( LDX, * )

PURPOSE
DGBSVX uses the LU factorization to compute the solution to a real sys‐
tem of linear equations A * X = B, A**T * X = B, or A**H * X = B, where
A is a band matrix of order N with KL subdiagonals and  KU  superdiago‐
nals, and X and B are N-by-NRHS matrices.
Error  bounds  on  the  solution and a condition estimate are also pro‐
vided.

DESCRIPTION
The following steps are performed by this subroutine:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N':  diag(R)*A*diag(C)	 *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = L * U,
where L is a product of permutation and unit lower triangular
matrices with KL subdiagonals, and U is upper triangular with
KL+KU superdiagonals.
3. If some U(i,i)=0, so that U is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A.  If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.  4.  The
system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.

ARGUMENTS
FACT    (input) CHARACTER*1
Specifies  whether  or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A  should  be
equilibrated  before it is factored.  = 'F':  On entry, AFB and
IPIV contain the factored form of A.  If EQUED is not 'N',  the
matrix  A has been equilibrated with scaling factors given by R
and C.  AB, AFB, and IPIV are not modified.  = 'N':  The matrix
A will be copied to AFB and factored.
=  'E':	 The  matrix A will be equilibrated if necessary, then
copied to AFB and factored.

TRANS   (input) CHARACTER*1
Specifies the form of the system of equations.  = 'N':  A * X =
B     (No transpose)
= 'T':  A**T * X = B  (Transpose)
= 'C':  A**H * X = B  (Transpose)

N       (input) INTEGER
The  number  of linear equations, i.e., the order of the matrix
A.  N >= 0.

KL      (input) INTEGER
The number of subdiagonals within the band of A.	 KL >= 0.

KU      (input) INTEGER
The number of superdiagonals within the band of A.  KU >= 0.

NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of columns  of
the matrices B and X.  NRHS >= 0.

AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
On  entry,  the matrix A in band storage, in rows 1 to KL+KU+1.
The j-th column of A is stored in the j-th column of the	 array
AB   as	 follows:   AB(KU+1+i-j,j)   =	 A(i,j)	 for  max(1,j-
KU)<=i<=min(N,j+kl) If FACT = 'F' and EQUED is not 'N', then  A
must  have been equilibrated by the scaling factors in R and/or
C.  AB is not modified if FACT = 'F' or 'N', or if FACT	=  'E'
and  EQUED  =  'N'  on  exit.  On exit, if EQUED .ne. 'N', A is
scaled as follows: EQUED = 'R':	A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and on entry  con‐
tains  details of the LU factorization of the band matrix A, as
computed by DGBTRF.  U is stored as an  upper  triangular  band
matrix  with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
multipliers used during the factorization are  stored  in  rows
KL+KU+2	to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is the fac‐
tored form of the equilibrated matrix A.	 If FACT =  'N',  then
AFB is an output argument and on exit returns details of the LU
factorization of A.  If FACT = 'E', then AFB is an output argu‐
ment and on exit returns details of the LU factorization of the
equilibrated matrix A (see the description of AB for  the  form
of the equilibrated matrix).

LDAFB   (input) INTEGER
The leading dimension of the array AFB.	LDAFB >= 2*KL+KU+1.

IPIV    (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry con‐
tains the pivot indices from the factorization A = L*U as  com‐
puted  by DGBTRF; row i of the matrix was interchanged with row
IPIV(i).	 If FACT = 'N', then IPIV is an output argument and on
exit  contains the pivot indices from the factorization A = L*U
of the original matrix A.  If FACT = 'E', then IPIV is an  out‐
put  argument  and  on exit contains the pivot indices from the
factorization A = L*U of the equilibrated matrix A.

EQUED   (input or output) CHARACTER*1
Specifies the form of equilibration that was done.  = 'N':   No
equilibration (always true if FACT = 'N').
=  'R':	 Row  equilibration, i.e., A has been premultiplied by
diag(R).	 = 'C':	 Column equilibration, i.e., A has been	 post‐
multiplied  by diag(C).	= 'B':	Both row and column equilibra‐
tion, i.e., A has been replaced	by  diag(R)  *	A  *  diag(C).
EQUED  is  an input argument if FACT = 'F'; otherwise, it is an
output argument.

R       (input or output) DOUBLE PRECISION array, dimension (N)
The row scale factors for A.  If EQUED = 'R' or 'B', A is  mul‐
tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
accessed.  R is an input argument if FACT = 'F';	 otherwise,  R
is  an  output argument.	 If FACT = 'F' and EQUED = 'R' or 'B',
each element of R must be positive.

C       (input or output) DOUBLE PRECISION array, dimension (N)
The column scale factors for A.	If EQUED = 'C' or  'B',	 A  is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
not accessed.  C is an input argument if FACT = 'F'; otherwise,
C is an output argument.	 If FACT = 'F' and EQUED = 'C' or 'B',
each element of C must be positive.

B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.	On exit,  if  EQUED  =
'N',  B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B',
B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and	 EQUED
= 'C' or 'B', B is overwritten by diag(C)*B.

LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).

X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If  INFO	 = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
the original system of equations.  Note that A and B are	 modi‐
fied on exit if EQUED .ne. 'N', and the solution to the equili‐
brated system is inv(diag(C))*X if TRANS = 'N' and EQUED =  'C'
or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R'
or 'B'.

LDX     (input) INTEGER
The leading dimension of the array X.  LDX >= max(1,N).

RCOND   (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix A
after  equilibration  (if  done).   If  RCOND  is less than the
machine precision (in particular, if RCOND = 0), the matrix  is
singular	 to working precision.	This condition is indicated by
a return code of INFO > 0.

FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the  j-th  column  of the solution matrix X).  If XTRUE is the
true solution corresponding to X(j), FERR(j)  is	 an  estimated
upper bound for the magnitude of the largest element in (X(j) -
XTRUE) divided by the magnitude of the largest element in X(j).
The  estimate  is as reliable as the estimate for RCOND, and is
almost always a slight overestimate of the true error.

BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution vec‐
tor  X(j) (i.e., the smallest relative change in any element of
A or B that makes X(j) an exact solution).

WORK    (workspace/output) DOUBLE PRECISION array, dimension (3*N)
On exit, WORK(1) contains the reciprocal	 pivot	growth	factor
norm(A)/norm(U).	 The  "max  absolute element" norm is used. If
WORK(1) is much less than 1, then the stability of the LU  fac‐
torization  of  the (equilibrated) matrix A could be poor. This
also means that the solution X, condition estimator RCOND,  and
forward	error bound FERR could be unreliable. If factorization
fails with 0<INFO<=N,  then  WORK(1)  contains  the  reciprocal
pivot growth factor for the leading INFO columns of A.

IWORK   (workspace) INTEGER array, dimension (N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, and i is
<= N:  U(i,i) is exactly zero.  The factorization has been com‐
pleted, but the factor U is exactly singular, so	 the  solution
and  error bounds could not be computed. RCOND = 0 is returned.
= N+1: U is nonsingular, but RCOND is less than machine	preci‐
sion, meaning that the matrix is singular to working precision.
Nevertheless,  the  solution  and  error	 bounds	 are  computed
because	there  are  a  number of situations where the computed
solution can be more accurate than the  value  of  RCOND	 would
suggest.

LAPACK driver routine (version 3November 2008			     DGBSVX(1)
```
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