SPBSVX(3F) SPBSVX(3F)
SPBSVX - use 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,
SUBROUTINE SPBSVX( 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
REAL RCOND
INTEGER IWORK( * )
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( *
), FERR( * ), S( * ), WORK( * ), X( LDX, * )
SPBSVX 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, 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 provided.
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. 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, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
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SPBSVX(3F) SPBSVX(3F)
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.
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
contains 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) REAL 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 equilibrated
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).
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SPBSVX(3F) SPBSVX(3F)
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >= KD+1.
AFB (input or output) REAL array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization 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 triangular 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 equilibrated 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) REAL 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 argument.
If FACT = 'F' and EQUED = 'Y', each element of S must be
positive.
B (input/output) REAL 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 overwritten
by diag(S) * B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0, 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 equilibrated system is
inv(diag(S))*X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
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SPBSVX(3F) SPBSVX(3F)
RCOND (output) REAL
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, and the solution and error bounds are not
computed.
FERR (output) REAL 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) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
that makes X(j) an exact solution).
WORK (workspace) REAL 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 definite,
so the factorization could not be completed, and the solution has
not been computed. = N+1: RCOND is less than machine precision.
The factorization has been completed, but the matrix is singular
to working precision, and the solution and error bounds have not
been computed.
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:
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SPBSVX(3F) SPBSVX(3F)
* * 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.
SPBSVX(3F) SPBSVX(3F)
SPBSVX - use 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,
SUBROUTINE SPBSVX( 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
REAL RCOND
INTEGER IWORK( * )
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( *
), FERR( * ), S( * ), WORK( * ), X( LDX, * )
SPBSVX 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, 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 provided.
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. 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, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
Page 1
SPBSVX(3F) SPBSVX(3F)
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.
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
contains 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) REAL 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 equilibrated
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).
Page 2
SPBSVX(3F) SPBSVX(3F)
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >= KD+1.
AFB (input or output) REAL array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization 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 triangular 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 equilibrated 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) REAL 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 argument.
If FACT = 'F' and EQUED = 'Y', each element of S must be
positive.
B (input/output) REAL 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 overwritten
by diag(S) * B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0, 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 equilibrated system is
inv(diag(S))*X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Page 3
SPBSVX(3F) SPBSVX(3F)
RCOND (output) REAL
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, and the solution and error bounds are not
computed.
FERR (output) REAL 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) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vector
X(j) (i.e., the smallest relative change in any element of A or B
that makes X(j) an exact solution).
WORK (workspace) REAL 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 definite,
so the factorization could not be completed, and the solution has
not been computed. = N+1: RCOND is less than machine precision.
The factorization has been completed, but the matrix is singular
to working precision, and the solution and error bounds have not
been computed.
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:
Page 4
SPBSVX(3F) SPBSVX(3F)
* * 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.
PPPPaaaaggggeeee 5555 [ Back ]
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