hpsv#

Functions

void chpsv(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c64*  restrict AP,
          INT*  restrict ipiv,
          c64*  restrict B,
    const INT            ldb,
          INT*           info
);
void chpsv(const char *uplo, const INT n, const INT nrhs, c64 *restrict AP, INT *restrict ipiv, c64 *restrict B, const INT ldb, INT *info)#

CHPSV computes the solution to a complex system of linear equations. 

A * X = B,

where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as 

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

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

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
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. 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. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by chptrf, stored as a packed triangular matrix in the same storage format as A.

out
ipiv

Array of dimension n. The pivot indices from chptrf (0-based).

inout
B

Array of dimension (ldb,nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,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, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

Functions

void zhpsv(
    const char*          uplo,
    const INT            n,
    const INT            nrhs,
          c128* restrict AP,
          INT*  restrict ipiv,
          c128* restrict B,
    const INT            ldb,
          INT*           info
);
void zhpsv(const char *uplo, const INT n, const INT nrhs, c128 *restrict AP, INT *restrict ipiv, c128 *restrict B, const INT ldb, INT *info)#

ZHPSV computes the solution to a complex system of linear equations. 

A * X = B,

where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as 

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

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

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
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. 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. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by zhptrf, stored as a packed triangular matrix in the same storage format as A.

out
ipiv

Array of dimension n. The pivot indices from zhptrf (0-based).

inout
B

Array of dimension (ldb,nrhs). On entry, the n-by-nrhs right hand side matrix B. On exit, if info=0, the n-by-nrhs solution matrix X.

in
ldb

The leading dimension of the array B. ldb>=max(1,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, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.