ppsv(3)

ppsv(3) — straight from the Library Functions Manual — is your bestie for factor and solve, and we're about to make it make sense 💖✨

🎀 NAME 🎀

ppsv — ppsv: factor and solve

✨ SYNOPSIS ✨

Okay so here are the functions, all lined up and looking iconic:

• cppsv — subroutine cppsv (uplo, n, nrhs, ap, b, ldb, info) CPPSV computes the solution to system of linear equations A * X = B for OTHER matrices
• dppsv — subroutine dppsv (uplo, n, nrhs, ap, b, ldb, info) DPPSV computes the solution to system of linear equations A * X = B for OTHER matrices
• sppsv — subroutine sppsv (uplo, n, nrhs, ap, b, ldb, info) SPPSV computes the solution to system of linear equations A * X = B for OTHER matrices
• zppsv — subroutine zppsv (uplo, n, nrhs, ap, b, ldb, info) ZPPSV computes the solution to system of linear equations A * X = B for OTHER matrices

🔍 Detailed Description 🔍

This is where the function docs live, and they are lowkey the whole point. 💅

🛠️ Function Documentation 🛠️

🌟 subroutine cppsv 🌟

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.

Parameters

• UPLO — UPLO is CHARACTER*1; here's the storage tea:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER; it's the number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER; it's 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, it's 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, it's 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, it's the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, it's the N-by-NRHS solution matrix X.
• LDB — LDB is INTEGER; it's the leading dimension of the array B. LDB >= max(1,N).
• INFO — INFO is INTEGER; status check, bestie:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the leading principal minor of order i of A is not positive, 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 ]

Definition at line 143 of file cppsv.f.

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🌟 subroutine dppsv 🌟

subroutine dppsv (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

DPPSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric 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**T* U,  if UPLO = 'U', or
A = L * L**T,  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.

Parameters

• UPLO — UPLO is CHARACTER*1; here's the storage tea:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER; it's the number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER; it's the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• AP — AP is DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, it's 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, if INFO = 0, it's the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A.

• B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, it's the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, it's the N-by-NRHS solution matrix X.
• LDB — LDB is INTEGER; it's the leading dimension of the array B. LDB >= max(1,N).
• INFO — INFO is INTEGER; status check, bestie:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the leading principal minor of order i of A is not positive, 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 symmetric 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 ]

Definition at line 143 of file dppsv.f.

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🌟 subroutine sppsv 🌟

subroutine sppsv (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

SPPSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric 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**T* U,  if UPLO = 'U', or
A = L * L**T,  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.

Parameters

• UPLO — UPLO is CHARACTER*1; here's the storage tea:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER; it's the number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER; it's the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• AP — AP is REAL array, dimension (N*(N+1)/2). On entry, it's 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, if INFO = 0, it's the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A.

• B — B is REAL array, dimension (LDB,NRHS). On entry, it's the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, it's the N-by-NRHS solution matrix X.
• LDB — LDB is INTEGER; it's the leading dimension of the array B. LDB >= max(1,N).
• INFO — INFO is INTEGER; status check, bestie:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the leading principal minor of order i of A is not positive, 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 ✨ (finishing sppsv)

Okay so the packed storage scheme is literally illustrated by this example when N = 4 and UPLO = 'U':

Two-dimensional storage of the symmetric 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 ]

And the official definition for this one lives at line 143 of file sppsv.f. Slay. 💅

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🦋 subroutine zppsv 🦋

subroutine zppsv (character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

So ZPPSV 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. Then the factored form of A is used to solve the system of equations A * X = B. Very demure, very Cholesky. 🎀

📋 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*16 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*16 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 principal minor of order i of A is not positive, 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 ✨ (zppsv edition)

The packed storage scheme is illustrated by the following example when N = 4 and 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 ]

Definition at line 143 of file zppsv.f. Iconic behavior, no notes. 🫶

🧚 Author (the Doxygen one)

Generated automatically by Doxygen for LAPACK from the source code.

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Okay that’s the whole ppsv(3) saga, bestie 💖 shoutout to LAPACK Version 3.12.0 and the ppsv(3) footer tag for carrying us through — you made it to the end, periodt. 🌈👑