ppsvx

sppsvx

Functions

void sppsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          f32*  restrict AP,
          f32*  restrict AFP,
          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 sppsvx(const char *fact, const char *uplo, const INT n, const INT nrhs, f32 *restrict AP, f32 *restrict AFP, 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)

SPPSVX 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 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='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 matrix and L is a lower triangular matrix.

3. 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.

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.

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

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:

a00 a01 a02 a03
    a11 a12 a13
        a22 a23     (aij = conjg(aji))
            a33

Packed storage of the upper triangle of A:

AP = [ a00, a01, a11, a02, a12, a22, a03, a13, a23, a33 ]

Parameters

in

fact

• 'F': AFP contains the factored form of A. If equed='Y', A has been equilibrated with scaling factors given by S; AP and AFP will not be modified.

• 'N': The matrix A will be copied to AFP and factored.

• 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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

nrhs

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

inout

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear 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 array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. See below for further details. A is not modified if fact='F' or 'N', or if fact='E' and equed='N' on exit. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

inout

AFP

Array of dimension n*(n+1)/2. If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization, in the same storage format as A. If fact='N' or 'E', an output argument returning the triangular factor of the (possibly equilibrated) matrix A.

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.

dppsvx

Functions

void dppsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          f64*  restrict AP,
          f64*  restrict AFP,
          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 dppsvx(const char *fact, const char *uplo, const INT n, const INT nrhs, f64 *restrict AP, f64 *restrict AFP, 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)

DPPSVX 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 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='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 matrix and L is a lower triangular matrix.

3. 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.

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.

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

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:

a00 a01 a02 a03
    a11 a12 a13
        a22 a23     (aij = conjg(aji))
            a33

Packed storage of the upper triangle of A:

AP = [ a00, a01, a11, a02, a12, a22, a03, a13, a23, a33 ]

Parameters

in

fact

• 'F': AFP contains the factored form of A. If equed='Y', A has been equilibrated with scaling factors given by S; AP and AFP will not be modified.

• 'N': The matrix A will be copied to AFP and factored.

• 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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

nrhs

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

inout

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear 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 array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. See below for further details. A is not modified if fact='F' or 'N', or if fact='E' and equed='N' on exit. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

inout

AFP

Array of dimension n*(n+1)/2. If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization, in the same storage format as A. If fact='N' or 'E', an output argument returning the triangular factor of the (possibly equilibrated) matrix A.

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.

cppsvx

Functions

void cppsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c64*  restrict AP,
          c64*  restrict AFP,
          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 cppsvx(const char *fact, const char *uplo, const INT n, const INT nrhs, c64 *restrict AP, c64 *restrict AFP, 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)

CPPSVX 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 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='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**H * U,  if uplo = 'U', or
   A = L * L**H,  if uplo = 'L',
   

   where U is an upper triangular matrix and L is a lower triangular matrix.

3. 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.

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.

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

Further Details:

The packed storage scheme is illustrated by the following example when n=4, uplo='U':

Two-dimensional storage of the Hermitian matrix A:

a00 a01 a02 a03
    a11 a12 a13
        a22 a23     (aij = conjg(aji))
            a33

Packed storage of the upper triangle of A:

AP = [ a00, a01, a11, a02, a12, a22, a03, a13, a23, a33 ]

Parameters

in

fact

• 'F': AFP contains the factored form of A. If equed='Y', A has been equilibrated with scaling factors given by S; AP and AFP will not be modified.

• 'N': The matrix A will be copied to AFP and factored.

• 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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

nrhs

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

inout

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear 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 array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. See below for further details. A is not modified if fact='F' or 'N', or if fact='E' and equed='N' on exit. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

inout

AFP

Array of dimension n*(n+1)/2. If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization, in the same storage format as A. If fact='N' or 'E', an output argument returning the triangular factor of the (possibly equilibrated) matrix A.

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.

zppsvx

Functions

void zppsvx(
    const char*          fact,
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c128* restrict AP,
          c128* restrict AFP,
          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 zppsvx(const char *fact, const char *uplo, const INT n, const INT nrhs, c128 *restrict AP, c128 *restrict AFP, 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)

ZPPSVX 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 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='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**H * U,  if uplo = 'U', or
   A = L * L**H,  if uplo = 'L',
   

   where U is an upper triangular matrix and L is a lower triangular matrix.

3. 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.

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.

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

Further Details:

The packed storage scheme is illustrated by the following example when n=4, uplo='U':

Two-dimensional storage of the Hermitian matrix A:

a00 a01 a02 a03
    a11 a12 a13
        a22 a23     (aij = conjg(aji))
            a33

Packed storage of the upper triangle of A:

AP = [ a00, a01, a11, a02, a12, a22, a03, a13, a23, a33 ]

Parameters

in

fact

• 'F': AFP contains the factored form of A. If equed='Y', A has been equilibrated with scaling factors given by S; AP and AFP will not be modified.

• 'N': The matrix A will be copied to AFP and factored.

• 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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

nrhs

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

inout

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear 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 array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. See below for further details. A is not modified if fact='F' or 'N', or if fact='E' and equed='N' on exit. On exit, if fact='E' and equed='Y', A is overwritten by diag(S)*A*diag(S).

inout

AFP

Array of dimension n*(n+1)/2. If fact='F', an input argument containing the triangular factor U or L from the Cholesky factorization, in the same storage format as A. If fact='N' or 'E', an output argument returning the triangular factor of the (possibly equilibrated) matrix A.

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.