laed1(3)

✨ NAME ✨

okay so bestie, this is laed1(3) from the Library Functions Manual — and the whole vibe is laed1 - laed1: D&C step: merge subproblems. That's the name, that's the mission, and yes she's a divide-and-conquer step that merges subproblems. periodt. 💅

🦋 SYNOPSIS 🦋

💖 Functions 💖

• subroutine dlaed1 (n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info) — DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. 💜
• 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. 💜

🌈 Detailed Description 🌈

This part is pretty much "the function docs are the moment," so the full breakdown is right below. No extra lore, just the two routines and all their details. 🫶

👑 Function Documentation 👑

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

🎯 Purpose 🎯

DLAED1 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. DLAED7 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 formula 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:

• 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.
• 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.
• 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 — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0. 🦋
• D — DOUBLE PRECISION array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix. 💖
• Q — 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 — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N). 🌟
• INDXQ — 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 — DOUBLE PRECISION. The subdiagonal entry used to create the rank-1 modification. 💅
• CUTPNT — INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2. 🫶
• WORK — DOUBLE PRECISION array, dimension (4*N + N**2). ✨
• IWORK — INTEGER array, dimension (4*N). ✨
• INFO — 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 📍

Definition at line 161 of file dlaed1.f.

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🎀 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 🎯

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. ✨

Same iconic formula, real-number edition:

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:

• 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 SLAED2.
• The second stage consists of 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.
• 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 — INTEGER. The dimension of the symmetric tridiagonal matrix. N >= 0. 🦋
• D — REAL array, dimension (N). On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix. 💖
• Q — 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 — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N). 🌟
• INDXQ — 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 — REAL. The subdiagonal entry used to create the rank-1 modification. 💅
• CUTPNT — INTEGER. The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2. 🫶
• WORK — REAL array, dimension (4*N + N**2). ✨
• IWORK — INTEGER array, dimension (4*N). ✨
• INFO — 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 📍

Definition at line 161 of file slaed1.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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catch u later, bestie 💖 — LAPACK Version 3.12.0, and that's the whole footer vibe (no date, just so u know 🫶).