DPBSVX(1) LAPACK driver routine (version 3.2) DPBSVX(1)NAME
DPBSVX - uses the Cholesky factorization A = U**T*U or A = L*L**T to
compute the solution to a real system of linear equations A * X = B,
SYNOPSIS
SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
IWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX,
* )
PURPOSE
DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to com‐
pute the solution to a real system of linear equations
A * X = B, where A is an N-by-N symmetric positive definite band
matrix 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:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular band matrix, and L is a lower
triangular band matrix.
3. If the leading i-by-i principal minor is not positive definite,
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(S) 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 con‐
tains the factored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S. AB and AFB
will not be 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.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of the matrix
A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array, except if
FACT = 'F' and EQUED = 'Y', then A must contain the equili‐
brated matrix diag(S)*A*diag(S). The j-th column of A is
stored in the j-th column of the array AB as follows: if UPLO =
'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; if UPLO =
'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). See below
for further details. On exit, if FACT = 'E' and EQUED = 'Y', A
is overwritten by diag(S)*A*diag(S).
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >= KD+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 the triangular factor U or L from the Cholesky factoriza‐
tion A = U**T*U or A = L*L**T of the band matrix A, in the same
storage format as A (see AB). If EQUED = 'Y', then AFB is the
factored form of the equilibrated matrix A. If FACT = 'N',
then AFB is an output argument and on exit returns the triangu‐
lar factor U or L from the Cholesky factorization A = U**T*U or
A = L*L**T. If FACT = 'E', then AFB is an output argument and
on exit returns the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T of the equilibrated
matrix A (see the description of A for the form of the equili‐
brated matrix).
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >= KD+1.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S). EQUED is an input argument if FACT =
'F'; otherwise, it is an output argument.
S (input or output) DOUBLE PRECISION array, dimension (N)
The scale factors for A; not accessed if EQUED = 'N'. S is an
input argument if FACT = 'F'; otherwise, S is an output argu‐
ment. If FACT = 'F' and EQUED = 'Y', each element of S must be
positive.
B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B. On exit, if
EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwrit‐
ten by diag(S) * 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 if EQUED = 'Y', A
and B are modified on exit, and the solution to the equili‐
brated system is inv(diag(S))*X.
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) DOUBLE PRECISION array, dimension (3*N)
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: the leading minor of order i of A is not positive defi‐
nite, so the factorization could not be completed, and the
solution has not been computed. RCOND = 0 is returned. = N+1:
U is nonsingular, but RCOND is less than machine precision,
meaning that the matrix is singular to working precision. Nev‐
ertheless, 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.
FURTHER DETAILS
The band storage scheme is illustrated by the following example, when N
= 6, KD = 2, and UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13
a22 a23 a24
a33 a34 a35
a44 a45 a46
a55 a56
(aij=conjg(aji)) a66
Band storage of the upper triangle of A:
* * a13 a24 a35 a46
* a12 a23 a34 a45 a56
a11 a22 a33 a44 a55 a66
Similarly, if UPLO = 'L' the format of A is as follows:
a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
a31 a42 a53 a64 * *
Array elements marked * are not used by the routine.
LAPACK driver routine (version 3November 2008 DPBSVX(1)
