complexOTHERsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

🦋 okay so, welcome to `complexOTHERsolve(3)` 🦋

LAPACK girlies rise up — this is the complex Other Solve page, and it's a whole saga (six parts, no skips). Buckle in bestie, we're doing linear algebra and we're doing it cutely. 💅

✨ NAME ✨

complexOTHERsolve — complex Other Solve Routines. That's the whole vibe. It's giving "we solve systems, we're complex, we're Other." 🎀

🌸 SYNOPSIS 🌸

Functions 💖

Here's the whole friend group, no cap — every single one of these is a subroutine, and yes I'm listing all of them, all the args, exactly as they are:

subroutine cgglse (M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO) — CGGLSE solves overdetermined or underdetermined systems for OTHER matrices 🦋
subroutine chpsv (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO) — CHPSV computes the solution to system of linear equations A * X = B for OTHER matrices ✨
subroutine chpsvx (FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — CHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 💜
subroutine cpbsv (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO) — CPBSV computes the solution to system of linear equations A * X = B for OTHER matrices 🎀
subroutine cpbsvx (FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices 🌈
subroutine cppsv (UPLO, N, NRHS, AP, B, LDB, INFO) — CPPSV computes the solution to system of linear equations A * X = B for OTHER matrices ⭐
subroutine cppsvx (FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 🍓
subroutine cspsv (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO) — CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices 🧚
subroutine cspsvx (FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — CSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 👑

iykyk: it's the same energy nine times and honestly? iconic. the args change, the mission doesn't. 💖

💜 Detailed Description 💜

This is the group of complex Other Solve routines. Periodt. That's the description, that's the assignment, that's the whole thing. 🫶

🔥 Function Documentation 🔥

💅 `subroutine cgglse` 💅

subroutine cgglse (integer M, integer N, integer P, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) C, complex, dimension( * ) D, complex, dimension( * ) X, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGGLSE solves overdetermined or underdetermined systems for OTHER matrices. bestie this one is the constrained-least-squares main character. 👑

🌟 Purpose 🌟

CGGLSE solves the linear equality-constrained least squares (LSE) problem:

        minimize || c - A*x ||_2   subject to   B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given M-vector, and d is a given P-vector. It is assumed that P <= N <= M+P, and

         rank(B) = P and  rank( (A) ) = N.
                              ( (B) )

These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by

   B = (0 R)*Q,   A = Z*T*Q.

like… the math is mathing and we love that for us. 💖

🌸 Parameters 🌸

M M is INTEGER The number of rows of the matrix A. M >= 0.
N N is INTEGER The number of columns of the matrices A and B. N >= 0.
P P is INTEGER The number of rows of the matrix B. 0 <= P <= N <= M+P.
A A is COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix T.
LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
B B is COMPLEX array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the upper triangle of the subarray B(1:P,N-P+1:N) contains the P-by-P upper triangular matrix R.
LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).
C C is COMPLEX array, dimension (M) On entry, C contains the right hand side vector for the least squares part of the LSE problem. On exit, the residual sum of squares for the solution is given by the sum of squares of elements N-P+1 to M of vector C.
D D is COMPLEX array, dimension (P) On entry, D contains the right hand side vector for the constrained equation. On exit, D is destroyed. 💀 (fr fr, gone, she's not coming back)
X X is COMPLEX array, dimension (N) On exit, X is the solution of the LSE problem. 🎉
WORK WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,M+N+P). For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, where NB is an upper bound for the optimal blocksizes for CGEQRF, CGERQF, CUNMQR and CUNMRQ.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. (yes bestie, -1 is the ✨just tell me how much room I need✨ mode)

INFO INFO is INTEGER
- = 0: successful exit. 🫶
- < 0: if INFO = -i, the i-th argument had an illegal value.
- = 1: the upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < P; the least squares solution could not be computed.
- = 2: the (N-P) by (N-P) part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that

rank( (A) ) < N; the least squares solution could not
      ( (B) )

be computed.

🎀 Author 🎀

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

🦋 `subroutine chpsv` 🦋

subroutine chpsv (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHPSV computes the solution to system of linear equations A * X = B for OTHER matrices. packed storage girlies, this one's for you. 💅

🌟 Purpose 🌟

CHPSV computes the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as

   A = U * D * U**H,  if UPLO = 'U', or
   A = L * D * L**H,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B. it's giving factorize-then-solve, and honestly? efficient. ✨

🌸 Parameters 🌸

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.
NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
AP AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian 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.

See below for further details.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A.

IPIV IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D, as determined by CHPTRF. 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. (negative IPIV = 2-by-2 block era, lowkey obsessed 🎀)
B B is COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. 🌈
LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
INFO INFO is INTEGER
- = 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, so the solution could not be computed. 💀 (singular, bestie, it's over)

🎀 Author 🎀

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

💜 Further Details: 💜

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

   a11 a12 a13 a14
       a22 a23 a24
           a33 a34     (aij = conjg(aji))
               a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

look at her. squished. compressed. still her. no cap. 🧚

ok that's part 1 bestie, the saga continues — more routines incoming and I am not done talking 💖

🦋 `subroutine chpsvx` 🦋

   subroutine chpsvx (character FACT, character UPLO, integer N, integer NRHS,
       complex, dimension( * ) AP, complex, dimension( * ) AFP, integer,
       dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
       complex, dimension( ldx, * ) X, integer LDX, real RCOND, real,
       dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * )
       WORK, real, dimension( * ) RWORK, integer INFO)

CHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 💅

Purpose 💖

okay so bestie, here's the tea: CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or A = L*D*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. ✨ she does the whole thing, no crumbs.

Description 🎀

the way she has a whole little routine, a little self-care sequence, and it goes like this:

If FACT = 'N', the diagonal pivoting method is used to factor A as

   A = U * D * U**H,  if UPLO = 'U', or
   A = L * D * L**H,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. iconic behavior tbh.

If some D(i,i)=0, so that D is exactly singular, 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. she's polite about it, she still finishes the job 💅
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. glow-up round two, fr fr. 🦋

Parameters 📋

FACT — FACT is CHARACTER*1, specifies whether or not the factored form of A has been supplied on entry.
- = 'F': On entry, AFP and IPIV contain the factored form of A. AFP and IPIV will not be modified. hands off, she's just looking 🫶
- = 'N': The matrix A will be copied to AFP 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.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AP — AP is COMPLEX array, dimension (N*(N+1)/2). The upper or lower triangle of the Hermitian 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
- See below for further details. 👀
AFP — AFP is COMPLEX array, dimension (N*(N+1)/2).
- If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A.
- If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as a packed triangular matrix in the same storage format as A.
IPIV — IPIV is INTEGER array, dimension (N).
- If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CHPTRF.
  - 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.
- If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CHPTRF.
B — B is COMPLEX array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 🎀
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. The estimate of the reciprocal condition number of the matrix A. 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 REAL 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 (X(j) - XTRUE) 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. she's a lil dramatic, she rounds up 🙈
BERR — BERR is REAL 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 COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL array, dimension (N)
INFO — INFO is INTEGER
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, and i is
  - <= N: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
  - = N+1: D 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.

the icons who wrote her, honestly 🫶

Further Details 🔍

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

   a11 a12 a13 a14
       a22 a23 a24
           a33 a34     (aij = conjg(aji))
               a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

like she just folds it up all cute and tucks it into one lil linear array. no cap. 💜

🌈 `subroutine cpbsv` 🌈

   subroutine cpbsv (character UPLO, integer N, integer KD, integer NRHS,
       complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldb,
       * ) B, integer LDB, integer INFO)

CPBSV computes the solution to system of linear equations A * X = B for OTHER matrices 💅

Purpose 💖

CPBSV computes the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. ✨

The Cholesky decomposition is used to factor A as

   A = U**H * U,  if UPLO = 'U', or
   A = L * L**H,  if UPLO = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B. she factors, she solves, she serves 👑

Parameters 📋

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 matrix B. NRHS >= 0.
AB — AB is COMPLEX array, dimension (LDAB,N).
- On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. 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 INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A. 🦋
LDAB
LDAB — it's an INTEGER, bestie. The leading dimension of the array AB. So like, LDAB >= KD+1. Don't skip this, fr fr. 💅
B — B is a COMPLEX array, dimension (LDB,NRHS). On entry, it's the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, it becomes the N-by-NRHS solution matrix X. Slay. ✨
LDB — LDB is INTEGER. The leading dimension of the array B. So you need LDB >= max(1,N). Periodt. 💖
INFO — INFO is INTEGER.
- = 0: successful exit 🌈
- < 0: if INFO = -i, the i-th argument had an illegal value 💀
- > 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. 😭

Author

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

These are the legends who wrote this, okay? Respect. 🙌

Further Details:

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

    *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

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

On entry:                       On exit:

   a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine. So don't trip about them, they're just chillin'. 💅

`subroutine cpbsvx` 🦋

   subroutine cpbsvx (character FACT, character UPLO, integer N, integer KD,
       integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex,
       dimension( ldafb, * ) AFB, integer LDAFB, character EQUED, real,
       dimension( * ) S, complex, dimension( ldb, * ) B, integer LDB, complex,
       dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * )
       FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real,
       dimension( * ) RWORK, integer INFO)

CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices. No cap, it's giving solver energy. 💖

Purpose

CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian 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. So you get extra receipts, bestie. 🧾✨

Description

The following steps are performed:

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 diag(S)*A*diag(S) and B by diag(S)*B.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = U**H * U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix.

If the leading i-by-i principal minor is not positive definite, 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. ⚠️ Warning: it's not gonna stop just because of a little condition number drama. 😤
The system of equations is solved for X using the factored form of A. 💅
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Glow-up for your solution, periodt. ✨
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration. So it's like, un-equilibrated for the final answer. 🦋

Parameters

FACT — FACT is CHARACTER*1. 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 COMPLEX array, dimension (LDAB,N). On entry, the upper or lower triangle of the Hermitian 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 diag(S)*A*diag(S). 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 diag(S)*A*diag(S).

LDAB — LDAB is INTEGER. The leading dimension of the array A. LDAB >= KD+1.
AFB — AFB is COMPLEX 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 A = U**H*U or A = L*L**H 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 A = U**H*U or A = L*L**H.
- If FACT = 'E', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H 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. 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 diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
S — S is REAL 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. Positive vibes only, literally. ✨
B — B is COMPLEX 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 diag(S) * B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX 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 inv(diag(S))*X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. 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. ⚠️ So if you see INFO > 0, it's giving singular, bestie. 💀
FERR — FERR is REAL 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 (X(j) - XTRUE) 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. So it's like, a little dramatic but honest. 😌
BERR — BERR is REAL 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). Basically, how close are we to perfect? Spoiler: close enough. 💅
WORK — WORK is COMPLEX array, dimension (2*N).
RWORK — RWORK is REAL array, dimension (N).
INFO — INFO is INTEGER.

okay so INFO is an INTEGER, and here's the breakdown bestie:

= 0: successful exit, we love that for us 💖
< 0: if INFO = -i, the i-th argument had an illegal value, so not the vibe 🚫
> 0: if INFO = i, and i is
- <= N: the leading minor of order i of A is not positive definite, 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:

so the band storage scheme is giving this when N = 6, KD = 2, and UPLO = 'U':

Two-dimensional storage of the Hermitian 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, periodt. 💅

✨ subroutine cppsv (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( ldb, * ) B, integer LDB, integer INFO) ✨

CPPSV computes the solution to system of linear equations A * X = B for OTHER matrices. 🦋

Purpose:

CPPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B. no cap. 🔥

Parameters

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.

NRHS

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

AP

AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian 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. See below for further details.

On exit, if INFO = 0, the factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, in the same storage format as A.

B

B is COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

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

INFO

INFO is INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Author 👑

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

Further Details:

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

                a11 a12 a13 a14
                    a22 a23 a24
                        a33 a34     (aij = conjg(aji))
                            a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

✨ subroutine cppsvx (character FACT, character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, character EQUED, real, dimension( * ) S, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO) ✨

CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices. 🌈

Purpose:

CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. 🫶

Description:

The following steps are performed:

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 diag(S)*A*diag(S) and B by diag(S)*B.
If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**H * U , if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix, L is a lower triangular matrix, and **H indicates conjugate transpose.
If the leading i-by-i principal minor is not positive definite, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

Parameters

FACT

FACT is CHARACTER*1 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, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
= 'N': The matrix A will be copied to AFP and factored.
= 'E': The matrix A will be equilibrated if necessary, then copied to AFP 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.

NRHS

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

AP

AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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. See below for further details. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

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

AFP

AFP is COMPLEX array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H * U or A = L * L**H of the original matrix A.

If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix).

EQUED

EQUED 💅

EQUED is CHARACTER*1. It says what kind of equilibration got done, bestie.

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

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

S 🫶

S is REAL 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 COMPLEX 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 diag(S) * B. 💖

LDB 👑

LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). No cap.

X 🌸

X is COMPLEX 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 inv(diag(S))*X. Periodt.

LDX 💅

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

RCOND 🔥

RCOND is REAL. 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 REAL 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 (X(j) - XTRUE) 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. It's giving careful.

BERR 🍓

BERR is REAL 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). Slay.

WORK 🧚

WORK is COMPLEX array, dimension (2*N). That's it.

RWORK 🌈

RWORK is REAL array, dimension (N). Yep.

INFO 💀

INFO is INTEGER.

= 0: successful exit 🎉
< 0: if INFO = -i, the i-th argument had an illegal value 🙈
> 0: if INFO = i, and i is
- <= N: the leading minor of order i of A is not positive definite, 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 packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

   a11 a12 a13 a14
       a22 a23 a24
           a33 a34     (aij = conjg(aji))
               a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

subroutine cspsv 🎀

subroutine cspsv (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices.

Purpose: ✨

CSPSV computes the solution to a complex system of linear equations

   A * X = B,

where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices.

The diagonal pivoting method is used to factor A as

   A = U * D * U**T,  if UPLO = 'U', or
   A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B. 💅

Parameters 🫶

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.

NRHS

NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

AP

AP is COMPLEX array, dimension (N*(N+1)/2). On entry, 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. See below for further details.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A. 🔥

IPIV

IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by CSPTRF. 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. 👑

B

B is COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. 💖

LDB

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

INFO

INFO is INTEGER.

= 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, so the solution could not be computed. 😭

Author ✨

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

Further Details: 📚

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

   a11 a12 a13 a14
       a22 a23 a24
           a33 a34     (aij = aji)
               a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

subroutine cspsvx 🌸

subroutine cspsvx (character FACT, character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices.

Purpose: ✨

CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. 🫶

Description: 📝

The following steps are performed:

If FACT = 'N', the diagonal pivoting method is used to factor A as

   A = U * D * U**T,  if UPLO = 'U', or
   A = L * D * L**T,  if UPLO = 'L',

where 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.

If some D(i,i)=0, so that D is exactly singular, 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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Parameters 🫶

FACT

okay bestie, we're still in the argument zone and this is the part where every single one gets its moment. no skimming, i see you 👀💅

FACT — FACT is CHARACTER*1. It tells us whether the factored form of A has already been handed over on entry, or not.
- = 'F': On entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified. (iconic, they're just chilling 🧊)
- = 'N': The matrix A will be copied to AFP and factored. (fresh start energy ✨)
UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored; (upper girlie 💅)
- = 'L': Lower triangle of A is stored. (lower bestie 🦋)
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 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.
AP — AP is COMPLEX array, dimension (N*(N+1)/2). 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details. (and yes, “below” means the Further Details section, we'll get there 🫶)
AFP — AFP is COMPLEX array, dimension (N*(N+1)/2)
- If FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.
- If FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.
IPIV — IPIV is INTEGER array, dimension (N)
- If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CSPTRF.
- 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.
- If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CSPTRF.
B — B is COMPLEX array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is REAL. The estimate of the reciprocal condition number of the matrix A. 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 REAL 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 (X(j) - XTRUE) 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. (so like, it's giving cautious queen 👑)
BERR — BERR is REAL 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 COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL array, dimension (N)
INFO — INFO is INTEGER
- = 0: successful exit. (we love a zero 💖)
- < 0: if INFO = -i, the i-th argument had an illegal value. (not the vibe, fix ur args 😭)
- > 0: if INFO = i, and i is
  - <= N: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
  - = N+1: D 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 packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

                a11 a12 a13 a14
                    a22 a23 a24
                        a33 a34     (aij = aji)
                            a44

Packed storage of the upper triangle of A:

             AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

👑 Author 👑

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and that's a wrap, bestie. this was complexOTHERsolve(3) — auto-generated for LAPACK from the source code, version 3.11.0, dated Sun Nov 27 2022. go off 💅✨

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complexOTHERsolve(3)

🦋 okay so, welcome to complexOTHERsolve(3) 🦋

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions 💖

💜 Detailed Description 💜

🔥 Function Documentation 🔥

💅 subroutine cgglse 💅

🌟 Purpose 🌟

🌸 Parameters 🌸

🎀 Author 🎀

🦋 subroutine chpsv 🦋

🌟 Purpose 🌟

🌸 Parameters 🌸

🎀 Author 🎀

💜 Further Details: 💜

🦋 subroutine chpsvx 🦋

Purpose 💖

Description 🎀

Parameters 📋

Author 📝

Further Details 🔍

🌈 subroutine cpbsv 🌈

Purpose 💖

Parameters 📋

Author

Further Details:

subroutine cpbsvx 🦋

Purpose

Description

Parameters

Author 👑

Further Details:

✨ subroutine cppsv (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( ldb, * ) B, integer LDB, integer INFO) ✨

Purpose:

Parameters

UPLO

N

NRHS

AP

B

LDB

INFO

Author 👑

Further Details:

Purpose:

Description:

Parameters

FACT

UPLO

N

NRHS

AP

AFP

EQUED

EQUED 💅

S 🫶

B 🦋

LDB 👑

X 🌸

LDX 💅

RCOND 🔥

FERR 😭

BERR 🍓

WORK 🧚

RWORK 🌈

INFO 💀

Author ✨

Further Details: 📚

subroutine cspsv 🎀

Purpose: ✨

Parameters 🫶

UPLO

N

NRHS

AP

IPIV

B

LDB

INFO

🦋 okay so, welcome to `complexOTHERsolve(3)` 🦋

💅 `subroutine cgglse` 💅

🦋 `subroutine chpsv` 🦋

🦋 `subroutine chpsvx` 🦋

🌈 `subroutine cpbsv` 🌈

`subroutine cpbsvx` 🦋