zspsv (l) - Linux Manuals

zspsv: computes the solution to a complex system of linear equations A * X = B,

NAME

ZSPSV - computes the solution to a complex system of linear equations A * X = B,

SYNOPSIS

SUBROUTINE ZSPSV(
UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

    
CHARACTER UPLO

    
INTEGER INFO, LDB, N, NRHS

    
INTEGER IPIV( * )

    
COMPLEX*16 AP( * ), B( LDB, * )

PURPOSE

ZSPSV computes 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.
The diagonal pivoting method is used to factor A as

A = U * D * U**T,  if UPLO = aqUaq, or

A = L * D * L**T,  if UPLO = aqLaq,
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric 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.

ARGUMENTS

UPLO (input) CHARACTER*1
= aqUaq: Upper triangle of A is stored;
= aqLaq: 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 matrix B. NRHS >= 0.
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, 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 = aqUaq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = aqLaq, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 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**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 (output) INTEGER array, dimension (N)
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 = aqUaq and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = aqLaq 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.
B (input/output) COMPLEX*16 array, 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.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 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.

FURTHER DETAILS

The packed storage scheme is illustrated by the following example when N = 4, UPLO = aqUaq:
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 ]

Pages related to zspsv

  • zspsv (3)
  • zspsvx (l) - 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
  • zspcon (l) - estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF
  • zspmv (l) - performs the matrix-vector operation y := alpha*A*x + beta*y,
  • zspr (l) - performs the symmetric rank 1 operation A := alpha*x*conjg( xaq ) + A,
  • zsprfs (l) - improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution
  • zsptrf (l) - computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method
  • zsptri (l) - computes the inverse of a complex symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF
  • zsptrs (l) - solves a system of linear equations A*X = B with a complex symmetric matrix A stored in packed format using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF