DSPTRF(3)

💅 DSPTRF(3) — LAPACK Library Functions Manual 💖

💅 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsptrf.f — that's the file, bestie. This page is DSPTRF from the LAPACK Library Functions Manual, and it's giving real-symmetric-packed-factorization main character energy. ✨

---

✨ SYNOPSIS

🫶 Functions/Subroutines

subroutine DSPTRF (uplo, n, ap, ipiv, info)
           DSPTRF

🌸 Function/Subroutine Documentation

subroutine DSPTRF (character uplo, integer n, double precision, dimension( *
       ) ap, integer, dimension( * ) ipiv, integer info)
       DSPTRF

---

🎀 Purpose:

okay so DSPTRF computes the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T  or  A = L*D*L**T

Here U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The way that this is literally the whole vibe of the routine? Obsessed. 🦋

---

🌸 Parameters

• UPLO — CHARACTER*1. This is the triangle vibe check:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — INTEGER. The order of the matrix A. N >= 0.
• AP — DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, this is 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 = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
  • if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L are stored as a packed triangular matrix overwriting A (see below for further details). Not me getting emotional over an overwrite. 😭

• IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D:
  • 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.
• INFO — INTEGER. The exit report card:
  • = 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, and division by zero will occur if it is used to solve a system of equations.

⚠️ Danger, bestie: when INFO = i > 0, D(i,i) is exactly zero. The factorization finished, but D is exactly singular, and dividing by zero will happen if you use it to solve a system of equations. That is not a vibe. 💀

---

💖 Author

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

---

🔥 Further Details:

If UPLO = 'U', then A = U*D*U**T, where

  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

            (   I    v    0   )   k-s
    U(k) =  (   0    I    0   )   s
            (   0    0    I   )   n-k
               k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

            (   I    0     0   )  k-1
    L(k) =  (   0    I     0   )  s
            (   0    v     I   )  n-k-s+1
               k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

---

🦋 Contributors:

J. Lewis, Boeing Computer Services Company — iconic contribution, periodt. 👑

---

📌 Definition at line 158 of file dsptrf.f.

fr fr, that's where the source lives.

---

🧚 Author

Generated automatically by Doxygen for LAPACK from the source code.

---

💖 That's the whole page, bestie! LAPACK Version 3.12.0 — footer path /home/abui...RC/dsptrf.f(3) — slay and stay packed. 🎀