Derived-types for matrix factorizations : QR

[loiseaujc]

Following #1040, this PR proposes a derived-type to store the QR factorization of a matrix. It is quite similar to what is being done in Julia and use xGEQRT driver from lapack to store the QR factorization using the compact WY format which is beneficial for performances (blocked algorithm + efficient specialized drivers matrix-matrix products).

Proposed interfaces

use stdlib_linalg, only: qr_rdp_type, qrfact
type(qr_rdp_type) :: F

! QR factorization of the matrix.
F = qrfact(A)

! Extract the Q and R components.
R = F%R()
Q = F%Q()

I still need to iterate a bit for the matrix-vector/matrix-matrix product.

Note - I don't quite like the qr_rdp_type naming for the derived-type but I ain't sure how to come with a better name. An alternative would be to use parameterized types but I'm not too confident about the actual support by all the compilers we test in the CI.

Other potential additions

This may be the subject of a different PR but I'll start the discussion here anyway as it may be relevant. Once the implementation is working/validated/integrated, I was wondering whether we should eventually defined an equivalent of the \ operator in Matlab and Julia for solving linear systems with a very simple x = A \ b which dispatch to the appropriate driver (solve for square matrix, lstsq for overdetermined ones, solve_lu if A is coming from the future lufact, specialized one for qrfact, etc). This is essentially syntactic sugar coating I'll admit but it can be a nice selling point for new comers: don't worry too much about how to get the performances out of your code, we make sure you have very descent performances by default (but if you really want to get the best perfs possible, you still need to go low-level and avoid allocations, etc).

Progress

• Interfaces
  • qrfact : Instantiate derived-type for QR factorization.
  • qrmv/qrmm : Specialized driver for matrix-vector and matrix-matrix products.
• Auxiliary functions
  • handle_geqrt_info : Handle info flag returned by geqrt (compact WY QR factorization).
  • handle_gemqrt_info : Handle info flag return by gemqrt (matrix-matrix product with compact WY storage).
• Base implementation
  • qrfact
  • qrmv/qrmm
  • self%Q() : Extract the Q matrix from the Compact WY representation.
  • self%R() : Extract the R matrix from the Compact WY representation.
• Tests
  • Random tall matrices
    • qrfact, self%Q(), self%R()
    • qrmv/qrmm
  • Random wide matrices
    • qrfact, self%Q(), self%R()
    • qrmv/qrmm
  • Random tall rank-deficient matrices
    • qrfact, self%Q(), self%R()
    • qrmv/qrmm
• In-code documentation
  • qrfact, self%Q(), self%R()
  • qrmv/qrmm
• Specifications
  • qrfact
  • qrmv/qrmm
• Examples

Ping: @perazz, @jvdp1, @jalvesz

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 68.55%. Comparing base (0ede301) to head (cdc4a9c).

Additional details and impacted files

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  Coverage   68.55%   68.55%           
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  Files         396      396           
  Lines       12746    12746           
  Branches     1376     1376           
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  Hits         8738     8738           
  Misses       4008     4008           

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