WIP: Hilbert/Banach space structures (#27401)

* change vecnorm to norm; introduce opnorm for matrix norms

* fixed vecnorm -> norm in tests for IterativeEigenolvers, complex, offsetarray

* dot,vecdot -> inner; deprecate vecdot

* add dot(x,y)=inner(x,y)

* use dot/inner instead of vecdot in test/offsetarray.jl

* replace length with length(LinearIndices(...

* const dot = inner

* norm(x,p) uses norm(xi) instead of norm(xi,p)

* update docstrings of norm and inner

* fixed isapprox for UniformScaling

* rename inner to dot

* fix norm of RowVector and some docstrings as suggested by @Sacha0

* additional test in 'VecOrMat of Vectors'

* removed a test in 'VecOrMat of Vectors'

* change vecdot for sparse matrices to dot

* added news

* fixed typo in NEWS and test in givens.jl

Author: ranocha

NEWS.md

@@ -511,6 +511,13 @@ This section lists changes that do not have deprecation warnings.
     This change makes `@schedule` redundant with `@async`, so `@schedule` has been
     deprecated ([#27164]).
 
+ * `norm(A::AbstractMatrix, p=2)` computes no longer the operator/matrix norm but the `norm` of `A`
+   as for other iterables, i.e. as if it were a vector. Especially, `norm(A::AbstractMatrix)` is the
+   Frobenius norm. To compute the operator/matrix norm, use the new function `opnorm` ([#27401]).
+
+  * `dot(u, v)` now acts recursively. Instead of `sum(u[i]' * v[i] for i in ...)`, it computes
+    `sum(dot(u[i], v[i]) for i in ...)`, similarly to `vecdot` before ([#27401]).
+
 Library improvements
 --------------------
 
@@ -1274,6 +1281,8 @@ Deprecated or removed
 
   * The functions `eigs` and `svds` have been moved to the `Arpack.jl` package ([#27616]).
 
+  * `vecdot` and `vecnorm` are deprecated in favor of `dot` and `norm`, respectively ([#27401]).
+
 Command-line option changes
 ---------------------------
 
@@ -1610,3 +1619,4 @@ Command-line option changes
 [#27189]: https://github.com/JuliaLang/julia/issues/27189
 [#27212]: https://github.com/JuliaLang/julia/issues/27212
 [#27248]: https://github.com/JuliaLang/julia/issues/27248
+[#27401]: https://github.com/JuliaLang/julia/issues/27401

stdlib/LinearAlgebra/docs/src/index.md

@@ -299,7 +299,6 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 Base.:*(::AbstractMatrix, ::AbstractMatrix)
 Base.:\(::AbstractMatrix, ::AbstractVecOrMat)
 LinearAlgebra.dot
-LinearAlgebra.vecdot
 LinearAlgebra.cross
 LinearAlgebra.factorize
 LinearAlgebra.Diagonal
@@ -358,7 +357,7 @@ LinearAlgebra.diag
 LinearAlgebra.diagm
 LinearAlgebra.rank
 LinearAlgebra.norm
-LinearAlgebra.vecnorm
+LinearAlgebra.opnorm
 LinearAlgebra.normalize!
 LinearAlgebra.normalize
 LinearAlgebra.cond

stdlib/LinearAlgebra/src/LinearAlgebra.jl

@@ -126,6 +126,7 @@ export
     qr!,
     lq,
     lq!,
+    opnorm,
     rank,
     rdiv!,
     schur,
@@ -143,8 +144,6 @@ export
     triu,
     tril!,
     triu!,
-    vecdot,
-    vecnorm,
 
 # Operators
     \,

stdlib/LinearAlgebra/src/dense.jl

@@ -5,7 +5,7 @@
 ## BLAS cutoff threshold constants
 
 const SCAL_CUTOFF = 2048
-const DOT_CUTOFF = 128
+#TODO const DOT_CUTOFF = 128
 const ASUM_CUTOFF = 32
 const NRM2_CUTOFF = 32
 
@@ -137,11 +137,11 @@ function norm(x::StridedVector{T}, rx::Union{UnitRange{TI},AbstractRange{TI}}) w
     GC.@preserve x BLAS.nrm2(length(rx), pointer(x)+(first(rx)-1)*sizeof(T), step(rx))
 end
 
-vecnorm1(x::Union{Array{T},StridedVector{T}}) where {T<:BlasReal} =
-    length(x) < ASUM_CUTOFF ? generic_vecnorm1(x) : BLAS.asum(x)
+norm1(x::Union{Array{T},StridedVector{T}}) where {T<:BlasReal} =
+    length(x) < ASUM_CUTOFF ? generic_norm1(x) : BLAS.asum(x)
 
-vecnorm2(x::Union{Array{T},StridedVector{T}}) where {T<:BlasFloat} =
-    length(x) < NRM2_CUTOFF ? generic_vecnorm2(x) : BLAS.nrm2(x)
+norm2(x::Union{Array{T},StridedVector{T}}) where {T<:BlasFloat} =
+    length(x) < NRM2_CUTOFF ? generic_norm2(x) : BLAS.nrm2(x)
 
 """
     triu!(M, k::Integer)
@@ -509,7 +509,7 @@ function exp!(A::StridedMatrix{T}) where T<:BlasFloat
         return copytri!(parent(exp(Hermitian(A))), 'U', true)
     end
     ilo, ihi, scale = LAPACK.gebal!('B', A)    # modifies A
-    nA   = norm(A, 1)
+    nA   = opnorm(A, 1)
     Inn    = Matrix{T}(I, n, n)
     ## For sufficiently small nA, use lower order Padé-Approximations
     if (nA <= 2.1)
@@ -1370,8 +1370,8 @@ function cond(A::AbstractMatrix, p::Real=2)
     end
     throw(ArgumentError("p-norm must be 1, 2 or Inf, got $p"))
 end
-_cond1Inf(A::StridedMatrix{<:BlasFloat}, p::Real) = _cond1Inf(lu(A), p, norm(A, p))
-_cond1Inf(A::AbstractMatrix, p::Real)             = norm(A, p)*norm(inv(A), p)
+_cond1Inf(A::StridedMatrix{<:BlasFloat}, p::Real) = _cond1Inf(lu(A), p, opnorm(A, p))
+_cond1Inf(A::AbstractMatrix, p::Real)             = opnorm(A, p)*opnorm(inv(A), p)
 
 ## Lyapunov and Sylvester equation
 

stdlib/LinearAlgebra/src/deprecated.jl

@@ -6,6 +6,12 @@ using Base: @deprecate, depwarn
 
 @deprecate cond(F::LinearAlgebra.LU, p::Integer) cond(convert(AbstractArray, F), p)
 
+# deprecate vecnorm in favor of norm
+@deprecate vecnorm norm
+
+# deprecate vecdot in favor of dot
+@deprecate vecdot dot
+
 # PR #22188
 export cholfact, cholfact!
 @deprecate cholfact!(A::StridedMatrix, uplo::Symbol, ::Type{Val{false}}) cholesky!(Hermitian(A, uplo), Val(false))