SRC/chpsvx.f(3)

💅 NAME 💅

SRC/chpsvx.f

okay so bestie, this page lives in the Library Functions Manual and it is literally all about SRC/chpsvx.f(3) ✨ we are getting into chpsvx and we are keeping every single detail because that's the vibe 💖

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

       subroutine chpsvx (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx,
           rcond, ferr, berr, work, rwork, info)
            CHPSVX computes the solution to system of linear equations A * X =
           B for OTHER matrices

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🌸 Function/Subroutine Documentation 🌸

   subroutine chpsvx (character fact, character uplo, integer n, integer nrhs,
       complex, dimension( * ) ap, complex, dimension( * ) afp, integer,
       dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,
       complex, dimension( ldx, * ) x, integer ldx, real rcond, real,
       dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * )
       work, real, dimension( * ) rwork, integer info)
        CHPSVX computes the solution to system of linear equations A * X = B
       for OTHER matrices

🎀 Purpose 🎀

okay so CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or A = L*D*L**H to compute 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 🫶

it also provides error bounds on the solution and a condition estimate, because it's generous like that 💅

🦋 Description 🦋

the following steps are performed:

1. if FACT = 'N', 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 and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks ✨
2. if some D(i,i)=0, so that D is exactly singular, 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 😭
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 👑

💖 Parameters 💖

• FACT — FACT is 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. AFP and IPIV will not be modified.
  • = 'N': the matrix A will be copied to AFP and factored.
• UPLO — UPLO is CHARACTER*1
  • = 'U': upper triangle of A is stored;
  • = 'L': lower triangle of A is stored.
• N — N is INTEGER. the number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER. the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
• AP — AP is COMPLEX array, dimension (N*(N+1)/2). 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-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 — AFP is COMPLEX 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**H or A = L*D*L**H as computed by CHPTRF, 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**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A.

• IPIV — IPIV is 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 CHPTRF.

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(k-1: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 D, as determined by CHPTRF.

• B — B is COMPLEX array, dimension (LDB,NRHS). the N-by-NRHS right hand side matrix B.
• LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
• X — X is COMPLEX array, dimension (LDX,NRHS). if INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
• LDX — LDX is INTEGER. the leading dimension of the array X. LDX >= max(1,N).
• RCOND — RCOND is REAL. 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.
• FERR — FERR is REAL 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 — BERR is REAL 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 — WORK is COMPLEX array, dimension (2*N)
• RWORK — RWORK is REAL array, dimension (N)
• INFO — INFO is INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, and i is
    • <= N: D(i,i) is exactly zero. the factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
    • = N+1: D 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. 💀

👑 Author 👑

the author squad is literally:

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🌈 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:

   a11 a12 a13 a14
       a22 a23 a24
           a33 a34     (aij = conjg(aji))
               a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

📍 Definition at line 275 of file chpsvx.f.

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie 💖 — LAPACK Version 3.12.0, SRC/chpsvx.f(3), mwah 💅✨