SRC/slaed1.f(3)

🦋 SRC/slaed1.f(3) 🦋

okay bestie, this is the Library Functions Manual entry for SRC/slaed1.f, and it's giving rank-one update realness in the most iconic way. 💅

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🎀 NAME 🎀

SRC/slaed1.f — yes, that's the name, bestie. 💖

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

subroutine slaed1 (n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info)

SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. 🔥

💖 Function/Subroutine Documentation 💖

🧚 subroutine slaed1 🧚

   subroutine slaed1 (integer n, real, dimension( * ) d, real, dimension( ldq,
       * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer
       cutpnt, real, dimension( * ) work, integer, dimension( * ) iwork,
       integer info)

SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. 👑

🔥 Purpose 🔥

okay so here's the tea, bestie: SLAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired. 💅

The equation is literally:

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. First stage: 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 SLAED2. 🫶
2. Second stage: calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem. 🔥
3. Final stage: 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 REAL array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix. 🦋
• Q — Q is REAL 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 REAL. 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 REAL array, dimension (4*N + N2)**. 🍓
• 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 slaed1.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. 🫶

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✨ Cute sign-off: LAPACK Version 3.12.0, source SRC/slaed1.f(3), and no date in this footer — timeless icon behavior, periodt. 💖