ZSPSVX(3F) ZSPSVX(3F)
ZSPSVX - use the diagonal pivoting factorization A = U*D*U**T or A =
L*D*L**T to compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N symmetric matrix stored in packed format
and X and B are N-by-NRHS matrices
SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A =
L*D*L**T to compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N symmetric matrix stored in packed format
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 = 'N', the diagonal pivoting method is used to factor A as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
2. 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 3 and 4 are skipped.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
Page 1
ZSPSVX(3F) ZSPSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': On entry, AFP and IPIV contain the factored
form of A. AP, AFP and IPIV will not be modified. = 'N': The
matrix A will be copied to AFP 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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored in
the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) =
A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) =
A(i,j) for j<=i<=n. See below for further details.
AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry
contains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**T or A
= L*D*L**T as computed by ZSPTRF, stored as a packed triangular
matrix in the same storage format as A.
If FACT = 'N', then AFP is an output argument and on exit
contains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**T or A
= L*D*L**T as computed by ZSPTRF, stored as a packed triangular
matrix in the same storage format as A.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block structure of
D, as determined by ZSPTRF. If IPIV(k) > 0, then rows and
columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then
rows and columns k-1 and -IPIV(k) were interchanged and D(k1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k)
= IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block structure of
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ZSPSVX(3F) ZSPSVX(3F)
D, as determined by ZSPTRF.
B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix 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.
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) 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 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0 and <= N: if INFO = i, D(i,i) is exactly zero. The
factorization has been completed, but the block diagonal matrix D
is exactly singular, so the solution and error bounds could not
be computed. = N+1: the block diagonal matrix D is nonsingular,
but RCOND is less than machine precision. The factorization has
been completed, but the matrix is singular to working precision,
so the solution and error bounds have not been computed.
Page 3
ZSPSVX(3F) ZSPSVX(3F)
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when N
= 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
ZSPSVX(3F) ZSPSVX(3F)
ZSPSVX - use the diagonal pivoting factorization A = U*D*U**T or A =
L*D*L**T to compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N symmetric matrix stored in packed format
and X and B are N-by-NRHS matrices
SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A =
L*D*L**T to compute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N symmetric matrix stored in packed format
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 = 'N', the diagonal pivoting method is used to factor A as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
2. 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 3 and 4 are skipped.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
Page 1
ZSPSVX(3F) ZSPSVX(3F)
FACT (input) CHARACTER*1
Specifies whether or not the factored form of A has been supplied
on entry. = 'F': On entry, AFP and IPIV contain the factored
form of A. AP, AFP and IPIV will not be modified. = 'N': The
matrix A will be copied to AFP 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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored in
the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) =
A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) =
A(i,j) for j<=i<=n. See below for further details.
AFP (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and on entry
contains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**T or A
= L*D*L**T as computed by ZSPTRF, stored as a packed triangular
matrix in the same storage format as A.
If FACT = 'N', then AFP is an output argument and on exit
contains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**T or A
= L*D*L**T as computed by ZSPTRF, stored as a packed triangular
matrix in the same storage format as A.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry
contains details of the interchanges and the block structure of
D, as determined by ZSPTRF. If IPIV(k) > 0, then rows and
columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then
rows and columns k-1 and -IPIV(k) were interchanged and D(k1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k)
= IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If FACT = 'N', then IPIV is an output argument and on exit
contains details of the interchanges and the block structure of
Page 2
ZSPSVX(3F) ZSPSVX(3F)
D, as determined by ZSPTRF.
B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix 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.
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) 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 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) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0 and <= N: if INFO = i, D(i,i) is exactly zero. The
factorization has been completed, but the block diagonal matrix D
is exactly singular, so the solution and error bounds could not
be computed. = N+1: the block diagonal matrix D is nonsingular,
but RCOND is less than machine precision. The factorization has
been completed, but the matrix is singular to working precision,
so the solution and error bounds have not been computed.
Page 3
ZSPSVX(3F) ZSPSVX(3F)
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when N
= 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
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