pbsvx#

spbsvx

Functions

void spbsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          f32*  restrict AB,
    const INT            ldab,
          f32*  restrict AFB,
    const INT            ldafb,
          char*          equed,
          f32*  restrict S,
          f32*  restrict B,
    const INT            ldb,
          f32*  restrict X,
    const INT            ldx,
          f32*           rcond,
          f32*  restrict ferr,
          f32*  restrict berr,
          f32*  restrict work,
          INT*  restrict iwork,
          INT*           info
);

void spbsvx(const char *fact, const char *uplo, const INT n, const INT kd, const INT nrhs, f32 *restrict AB, const INT ldab, f32 *restrict AFB, const INT ldafb, char *equed, f32 *restrict S, f32 *restrict B, const INT ldb, f32 *restrict X, const INT ldx, f32 *rcond, f32 *restrict ferr, f32 *restrict berr, f32 *restrict work, INT *restrict iwork, INT *info)#

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:

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.
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.
If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

equed is an input argument if fact='F'; otherwise it is an output argument.

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:

a00  a01  a02
     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45
(aij=conjg(aji))         a55

Band storage of the upper triangle of A:

 *    *   a02  a13  a24  a35
 *   a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

Similarly, if uplo='L' the format of A is as follows:

a00  a11  a22  a33  a44  a55
a10  a21  a32  a43  a54   *
a20  a31  a42  a53   *    *

Array elements marked * are not used by the routine.

Parameters

in

fact

'F': AFB contains the factored form of A. If equed='Y', 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.

in

uplo

'U': Upper triangle of A is stored
'L': Lower triangle of A is stored

in

n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

nrhs

The number of right-hand sides. nrhs>=0.

inout

AB

Array of 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+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

inout

AFB

Array of dimension (ldafb, n). If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization 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' or 'E', an output argument returning the triangular factor U or L from the Cholesky factorization of the (possibly equilibrated) matrix A.

in

ldafb

The leading dimension of the array AFB. ldafb>=kd+1.

inout

equed

'N': No equilibration (always true if fact='N')
'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S)

inout

S

Array of dimension (n). The scale factors for A; not accessed if equed='N'. S is an input argument if fact='F'; otherwise it is an output argument. If fact='F' and equed='Y', each element of S must be positive.

inout

B

Array of 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.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

X

Array of 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 equilibrated system is inv(diag(S))*X.

in

ldx

The leading dimension of the array X. ldx>=max(1,n).

out

rcond

The estimate of the reciprocal condition number of the matrix A after equilibration (if done).

out

ferr

Array of dimension (nrhs). The estimated forward error bound for each solution vector.

out

berr

Array of dimension (nrhs). The componentwise relative backward error of each solution vector.

out

work

Array of dimension 3*n.

out

iwork

Array of dimension n.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: if info=i, and i<=n, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. rcond=0 is returned.
info=n+1: U is nonsingular, but rcond is less than machine precision, 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.

dpbsvx

Functions

void dpbsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          f64*  restrict AB,
    const INT            ldab,
          f64*  restrict AFB,
    const INT            ldafb,
          char*          equed,
          f64*  restrict S,
          f64*  restrict B,
    const INT            ldb,
          f64*  restrict X,
    const INT            ldx,
          f64*           rcond,
          f64*  restrict ferr,
          f64*  restrict berr,
          f64*  restrict work,
          INT*  restrict iwork,
          INT*           info
);

void dpbsvx(const char *fact, const char *uplo, const INT n, const INT kd, const INT nrhs, f64 *restrict AB, const INT ldab, f64 *restrict AFB, const INT ldafb, char *equed, f64 *restrict S, f64 *restrict B, const INT ldb, f64 *restrict X, const INT ldx, f64 *rcond, f64 *restrict ferr, f64 *restrict berr, f64 *restrict work, INT *restrict iwork, INT *info)#

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

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:

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.
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.
If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

equed is an input argument if fact='F'; otherwise it is an output argument.

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:

a00  a01  a02
     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45
(aij=conjg(aji))         a55

Band storage of the upper triangle of A:

 *    *   a02  a13  a24  a35
 *   a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

Similarly, if uplo='L' the format of A is as follows:

a00  a11  a22  a33  a44  a55
a10  a21  a32  a43  a54   *
a20  a31  a42  a53   *    *

Array elements marked * are not used by the routine.

Parameters

in

fact

'F': AFB contains the factored form of A. If equed='Y', 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.

in

uplo

'U': Upper triangle of A is stored
'L': Lower triangle of A is stored

in

n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

nrhs

The number of right-hand sides. nrhs>=0.

inout

AB

Array of 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+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

inout

AFB

Array of dimension (ldafb, n). If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization 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' or 'E', an output argument returning the triangular factor U or L from the Cholesky factorization of the (possibly equilibrated) matrix A.

in

ldafb

The leading dimension of the array AFB. ldafb>=kd+1.

inout

equed

'N': No equilibration (always true if fact='N')
'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S)

inout

S

Array of dimension (n). The scale factors for A; not accessed if equed='N'. S is an input argument if fact='F'; otherwise it is an output argument. If fact='F' and equed='Y', each element of S must be positive.

inout

B

Array of 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.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

X

Array of 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 equilibrated system is inv(diag(S))*X.

in

ldx

The leading dimension of the array X. ldx>=max(1,n).

out

rcond

The estimate of the reciprocal condition number of the matrix A after equilibration (if done).

out

ferr

Array of dimension (nrhs). The estimated forward error bound for each solution vector.

out

berr

Array of dimension (nrhs). The componentwise relative backward error of each solution vector.

out

work

Array of dimension 3*n.

out

iwork

Array of dimension n.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: if info=i, and i<=n, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. rcond=0 is returned.
info=n+1: U is nonsingular, but rcond is less than machine precision, 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.

cpbsvx

Functions

void cpbsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          c64*  restrict AB,
    const INT            ldab,
          c64*  restrict AFB,
    const INT            ldafb,
          char*          equed,
          f32*  restrict S,
          c64*  restrict B,
    const INT            ldb,
          c64*  restrict X,
    const INT            ldx,
          f32*           rcond,
          f32*  restrict ferr,
          f32*  restrict berr,
          c64*  restrict work,
          f32*  restrict rwork,
          INT*           info
);

void cpbsvx(const char *fact, const char *uplo, const INT n, const INT kd, const INT nrhs, c64 *restrict AB, const INT ldab, c64 *restrict AFB, const INT ldafb, char *equed, f32 *restrict S, c64 *restrict B, const INT ldb, c64 *restrict X, const INT ldx, f32 *rcond, f32 *restrict ferr, f32 *restrict berr, c64 *restrict work, f32 *restrict rwork, INT *info)#

CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations.

A * X = B

where A is an n-by-n Hermitian 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:

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.
If fact='N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if fact='E') as:
```
A = U**H * U,  if uplo = 'U', or
A = L * L**H,  if uplo = 'L',
```
where U is an upper triangular band matrix, and L is a lower triangular band matrix.
If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

equed is an input argument if fact='F'; otherwise it is an output argument.

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 Hermitian matrix A:

a00  a01  a02
     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45
(aij=conjg(aji))         a55

Band storage of the upper triangle of A:

 *    *   a02  a13  a24  a35
 *   a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

Similarly, if uplo='L' the format of A is as follows:

a00  a11  a22  a33  a44  a55
a10  a21  a32  a43  a54   *
a20  a31  a42  a53   *    *

Array elements marked * are not used by the routine.

Parameters

in

fact

'F': AFB contains the factored form of A. If equed='Y', 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.

in

uplo

'U': Upper triangle of A is stored
'L': Lower triangle of A is stored

in

n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

nrhs

The number of right-hand sides. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the Hermitian 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+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

inout

AFB

Array of dimension (ldafb, n). If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization 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' or 'E', an output argument returning the triangular factor U or L from the Cholesky factorization of the (possibly equilibrated) matrix A.

in

ldafb

The leading dimension of the array AFB. ldafb>=kd+1.

inout

equed

'N': No equilibration (always true if fact='N')
'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S)

inout

S

Array of dimension (n). The scale factors for A; not accessed if equed='N'. S is an input argument if fact='F'; otherwise it is an output argument. If fact='F' and equed='Y', each element of S must be positive.

inout

B

Array of 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.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

X

Array of 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 equilibrated system is inv(diag(S))*X.

in

ldx

The leading dimension of the array X. ldx>=max(1,n).

out

rcond

The estimate of the reciprocal condition number of the matrix A after equilibration (if done).

out

ferr

Array of dimension (nrhs). The estimated forward error bound for each solution vector.

out

berr

Array of dimension (nrhs). The componentwise relative backward error of each solution vector.

out

work

Complex array of dimension 2*n.

out

rwork

Array of dimension n.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: if info=i, and i<=n, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. rcond=0 is returned.
info=n+1: U is nonsingular, but rcond is less than machine precision, 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.

zpbsvx

Functions

void zpbsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const INT            nrhs,
          c128* restrict AB,
    const INT            ldab,
          c128* restrict AFB,
    const INT            ldafb,
          char*          equed,
          f64*  restrict S,
          c128* restrict B,
    const INT            ldb,
          c128* restrict X,
    const INT            ldx,
          f64*           rcond,
          f64*  restrict ferr,
          f64*  restrict berr,
          c128* restrict work,
          f64*  restrict rwork,
          INT*           info
);

void zpbsvx(const char *fact, const char *uplo, const INT n, const INT kd, const INT nrhs, c128 *restrict AB, const INT ldab, c128 *restrict AFB, const INT ldafb, char *equed, f64 *restrict S, c128 *restrict B, const INT ldb, c128 *restrict X, const INT ldx, f64 *rcond, f64 *restrict ferr, f64 *restrict berr, c128 *restrict work, f64 *restrict rwork, INT *info)#

ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations.

A * X = B

where A is an n-by-n Hermitian 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:

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.
If fact='N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if fact='E') as:
```
A = U**H * U,  if uplo = 'U', or
A = L * L**H,  if uplo = 'L',
```
where U is an upper triangular band matrix, and L is a lower triangular band matrix.
If the leading principal minor of order i is not positive, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

equed is an input argument if fact='F'; otherwise it is an output argument.

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 Hermitian matrix A:

a00  a01  a02
     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45
(aij=conjg(aji))         a55

Band storage of the upper triangle of A:

 *    *   a02  a13  a24  a35
 *   a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

Similarly, if uplo='L' the format of A is as follows:

a00  a11  a22  a33  a44  a55
a10  a21  a32  a43  a54   *
a20  a31  a42  a53   *    *

Array elements marked * are not used by the routine.

Parameters

in

fact

'F': AFB contains the factored form of A. If equed='Y', 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.

in

uplo

'U': Upper triangle of A is stored
'L': Lower triangle of A is stored

in

n

The number of linear equations, i.e., the order of the matrix A. n>=0.

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

nrhs

The number of right-hand sides. nrhs>=0.

inout

AB

Array of dimension (ldab, n). On entry, the upper or lower triangle of the Hermitian 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+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd). See below for further details. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

inout

AFB

Array of dimension (ldafb, n). If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization 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' or 'E', an output argument returning the triangular factor U or L from the Cholesky factorization of the (possibly equilibrated) matrix A.

in

ldafb

The leading dimension of the array AFB. ldafb>=kd+1.

inout

equed

'N': No equilibration (always true if fact='N')
'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S)

inout

S

Array of dimension (n). The scale factors for A; not accessed if equed='N'. S is an input argument if fact='F'; otherwise it is an output argument. If fact='F' and equed='Y', each element of S must be positive.

inout

B

Array of 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.

in

ldb

The leading dimension of the array B. ldb>=max(1,n).

out

X

Array of 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 equilibrated system is inv(diag(S))*X.

in

ldx

The leading dimension of the array X. ldx>=max(1,n).

out

rcond

The estimate of the reciprocal condition number of the matrix A after equilibration (if done).

out

ferr

Array of dimension (nrhs). The estimated forward error bound for each solution vector.

out

berr

Array of dimension (nrhs). The componentwise relative backward error of each solution vector.

out

work

Complex array of dimension 2*n.

out

rwork

Array of dimension n.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: if info=i, and i<=n, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. rcond=0 is returned.
info=n+1: U is nonsingular, but rcond is less than machine precision, 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.