SRC/dpbsvx.f(3)

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SRC/dpbsvx.f — Library Functions Manual 💖

✨ NAME ✨

SRC/dpbsvx.f — that's literally the name, bestie, periodt. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine dpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
    s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
     DPBSVX computes the solution to system of linear equations A * X =
    B for OTHER matrices

💖 Function/Subroutine Documentation 💖

Here's the full typed-out signature, so you can see every single argument in its natural habitat:

subroutine dpbsvx (character fact, character uplo, integer n, integer kd,
    integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab,
    double precision, dimension( ldafb, * ) afb, integer ldafb, character
    equed, double precision, dimension( * ) s, double precision, dimension(
    ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x,
    integer ldx, double precision rcond, double precision, dimension( * )
    ferr, double precision, dimension( * ) berr, double precision,
    dimension( * ) work, integer, dimension( * ) iwork, integer info)

DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices — yes, that's the whole vibe. 🫶

🎯 Purpose 🎯

okay so here's the tea: DPBSVX uses the Cholesky factorization <code>A = UT*U</code> or <code>A = L*LT</code> 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. Slay. ✨

🧮 Description 🧮

The following steps are performed, bestie — buckle up:

  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 <code>diag(S)*A*diag(S)</code> and B by <code>diag(S)*B</code>. 👑

  1. 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. 🦋

  1. 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. 😭
  2. The system of equations is solved for X using the factored form of A. 💫
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 🔥
  4. If equilibration was used, the matrix X is premultiplied by <code>diag(S)</code> so that it solves the original system before equilibration. 🌈

🎛️ Parameters 🎛️

FACT 💖

FACT is CHARACTER*1. It specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

  • = 'F': On entry, AFB contains the factored form of A. If EQUED = 'Y', the matrix 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.

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.

KD 🎀

KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 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.

AB 🔥

AB is DOUBLE PRECISION array, 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 <code>diag(S)*A*diag(S)</code>. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).

See below for further details. 👀

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by <code>diag(S)*A*diag(S)</code>.

LDAB 🍓

LDAB is INTEGER. The leading dimension of the array A. LDAB >= KD+1.

AFB 💀

AFB is DOUBLE PRECISION array, dimension (LDAFB,N).

If FACT = 'F', then AFB is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization <code>A = UT*U</code> or <code>A = L*LT</code> 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', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization <code>A = UT*U</code> or <code>A = L*LT</code>.

If FACT = 'E', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization <code>A = UT*U</code> or <code>A = L*LT</code> of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). ✨

LDAFB 🧚

LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= KD+1.

EQUED 🌈

EQUED is CHARACTER*1. It specifies the form of equilibration that was done.

  • = 'N': No equilibration (always true if FACT = 'N').
  • = 'Y': Equilibration was done, i.e., A has been replaced by <code>diag(S) * A * diag(S)</code>.

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. 💅

S 🌟

S is DOUBLE PRECISION array, dimension (N). The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. 🫶

B 🎀

B is DOUBLE PRECISION array, 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 <code>diag(S) * B</code>. 💖

LDB 🦋

LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).

X 👑

X is DOUBLE PRECISION array, 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 <code>inv(diag(S))*X</code>. ✨

LDX 💜

LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).

RCOND 💫

RCOND is DOUBLE PRECISION. The estimate of the reciprocal condition number of the matrix A after equilibration (if done). 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 DOUBLE PRECISION 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 <code>(X(j) - XTRUE)</code> 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N).

IWORK 🌟

IWORK is INTEGER array, dimension (N).

INFO 💀

(okay the INFO deets are literally in part 2 — not leaving you hanging, bestie)

INFO is INTEGER — the little status bestie, and here’s the tea ☕✨

  • INFO = 0 — successful exit. slay 💅
  • INFO < 0 — if INFO = -i, the i-th argument had an illegal value. 💀
  • INFO > 0 — if INFO = i, and i is
    • <= 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. 😭
    • = 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. 🫶

💖 Author

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

🌸 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:

     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45  a46
                         a55  a56
     (aij=conjg(aji))         a66

Band storage of the upper triangle of A:

      *    *   a13  a24  a35  a46
      *   a12  a23  a34  a45  a56
     a11  a22  a33  a44  a55  a66

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

     a11  a22  a33  a44  a55  a66
     a21  a32  a43  a54  a65   *
     a31  a42  a53  a64   *    *

Array elements marked * are not used by the routine. ✨

Definition at line 340 of file dpbsvx.f.

🎀 Author

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