SRC/dlaed1.f(3)

🌸 NAME 🌸

okay so bestie, the name on this page is literally SRC/dlaed1.f, and it lives in the Library Functions Manual. That's the whole NAME section — short, cute, iconic. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

• subroutine dlaed1 (n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info) — DLAED1 is the one DSTEDC uses. It computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. It's the move when the original matrix is tridiagonal.

💖 Function/Subroutine Documentation 💖

👑 subroutine dlaed1 👑

subroutine dlaed1 (integer n, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAED1 is used by DSTEDC. It computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix, and it's used when the original matrix is tridiagonal. 🦋

🍓 Purpose 🍓

Okay so here's the tea, bestie: DLAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is only for the eigenproblem that wants all eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles the case where eigenvalues only, or eigenvalues and eigenvectors, of a full symmetric matrix (which was reduced to tridiagonal form) are desired. ✨

The equation is giving main-character energy:

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**T*u, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. 💫

The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages:

1. The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2.
2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED3). This routine also calculates the eigenvectors of the current problem.
3. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.

🌸 Parameters 🌸

• N — N is INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0. 💖
• D — D is DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix. 🎀
• Q — Q is DOUBLE PRECISION array, dimension (LDQ,N). On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix. 🌈
• LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N). ⭐
• INDXQ — INDXQ is INTEGER array, dimension (N). On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. 🦋
• RHO — RHO is DOUBLE PRECISION. The subdiagonal entry used to create the rank-1 modification. 🔥
• CUTPNT — CUTPNT is INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2. 👑
• WORK — WORK is DOUBLE PRECISION array, dimension (4*N + N**2). 🧚
• IWORK — IWORK is INTEGER array, dimension (4*N). 💜
• INFO — INFO is INTEGER.
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: if INFO = 1, an eigenvalue did not converge. 😭

💜 Author 💜

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

🫶 Contributors: 🫶

• Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
• Modified by Francoise Tisseur, University of Tennessee

Definition at line 161 of file dlaed1.f. 📌

🌈 Author 🌈

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