diff --git a/deps/checksums/gmp b/deps/checksums/gmp
index db01480b953d0..da510e3dc2388 100644
--- a/deps/checksums/gmp
+++ b/deps/checksums/gmp
@@ -1,3 +1,5 @@
+gmp-6.2.1.tar.bz2/md5/28971fc21cf028042d4897f02fd355ea
+gmp-6.2.1.tar.bz2/sha512/8904334a3bcc5c896ececabc75cda9dec642e401fb5397c4992c4fabea5e962c9ce8bd44e8e4233c34e55c8010cc28db0545f5f750cbdbb5f00af538dc763be9
 GMP.v6.2.1+0.aarch64-apple-darwin.tar.gz/md5/e805c580078e4d6bcaeb6781cb6d56fa
 GMP.v6.2.1+0.aarch64-apple-darwin.tar.gz/sha512/62435e80f5fa0b67e2788c8bfc3681426add7a9b2853131bbebe890d1a2d9b54cebaea0860f6ddd0e93e1ae302baba39851d5f58a65acf0b2a9ea1226bb4eea4
 GMP.v6.2.1+0.aarch64-linux-gnu-cxx03.tar.gz/md5/5384d6ba6fd408bc71c2781b643cd59a
@@ -56,5 +58,3 @@ GMP.v6.2.1+0.x86_64-w64-mingw32-cxx03.tar.gz/md5/1499a265b438cf5169286c1830eb573
 GMP.v6.2.1+0.x86_64-w64-mingw32-cxx03.tar.gz/sha512/d2e6fe76abe0a0cb1a7445ea93cd5bd0bf9f729aec8df9c76d06a1f6f5e67cce442be69b66950eb33aa22cfda2e5a308f2bade64018a27bebfcb4b7a97e1d047
 GMP.v6.2.1+0.x86_64-w64-mingw32-cxx11.tar.gz/md5/fdb4187f617511d8eb19f67f8499a8d0
 GMP.v6.2.1+0.x86_64-w64-mingw32-cxx11.tar.gz/sha512/bb6d8ead1c20cffebc2271461d3787cfad794fee2b32e23583af6521c0667ed9107805268a996d23d6edcab9fe653e542a210cab07252f7713af0c23feb76fb3
-gmp-6.2.1.tar.bz2/md5/28971fc21cf028042d4897f02fd355ea
-gmp-6.2.1.tar.bz2/sha512/8904334a3bcc5c896ececabc75cda9dec642e401fb5397c4992c4fabea5e962c9ce8bd44e8e4233c34e55c8010cc28db0545f5f750cbdbb5f00af538dc763be9
diff --git a/deps/checksums/libgit2 b/deps/checksums/libgit2
index 0603b2ab9a662..6d836e4a4501c 100644
--- a/deps/checksums/libgit2
+++ b/deps/checksums/libgit2
@@ -1,3 +1,5 @@
+libgit2-7f4fa178629d559c037a1f72f79f79af9c1ef8ce.tar.gz/md5/c6a819fb0bf924df61e1595624a0988a
+libgit2-7f4fa178629d559c037a1f72f79f79af9c1ef8ce.tar.gz/sha512/3de9c042115b309dae3b8e0008edf2e762addd90a7bdb54b3cf634811271ab9dbfea35656650eb53a3faec73caf33ed199fb885ec21f611c79d909d9f4fe48c5
 LibGit2.v1.2.3+0.aarch64-apple-darwin.tar.gz/md5/0eb4d35fa6078de8da3dc79a420d5bc5
 LibGit2.v1.2.3+0.aarch64-apple-darwin.tar.gz/sha512/aff23e9d56827be54d8f515489eed92d9cb4d9a1b69c12bbee44fa1343ebc906b38b082c11e6ecb227d256de5cba19b2f3a03c022b88809a33c225c34db08f7c
 LibGit2.v1.2.3+0.aarch64-linux-gnu.tar.gz/md5/f2f31aa978e43aa68a0e73107067c7cd
@@ -30,5 +32,3 @@ LibGit2.v1.2.3+0.x86_64-unknown-freebsd.tar.gz/md5/e1623fd3f8f564085d47ec650a40e
 LibGit2.v1.2.3+0.x86_64-unknown-freebsd.tar.gz/sha512/295d55b78b21ef1c2ba471c8b5618b168dd633e986db9e1ec3e9630e352446ab18e8fd0992010b6afdd922463bb285bc45885a8b35a502d574553fe61c1f7b9f
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 LibGit2.v1.2.3+0.x86_64-w64-mingw32.tar.gz/sha512/21951f3bc902f30b8cc75c3af233aa7fe8457e412e7758d556bf71de149c7f2325a5c4c204a7a462cc6a61b3dcb90f0d25e684ffd8617b0a1505a1d31cf2f69a
-libgit2-7f4fa178629d559c037a1f72f79f79af9c1ef8ce.tar.gz/md5/c6a819fb0bf924df61e1595624a0988a
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diff --git a/deps/checksums/openblas b/deps/checksums/openblas
index 722bd059bfd85..05b7ab29f55b7 100644
--- a/deps/checksums/openblas
+++ b/deps/checksums/openblas
@@ -1,94 +1,94 @@
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+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran3.tar.gz/md5/46114a6da46772f5a13330d1c4786f57
+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran3.tar.gz/sha512/a805be012462ed47bf153cbb7f2fe476e9988f57165927e4c6ce5e344c6580a6547f492d0df555e4f4f34c4a55efb179c32451f0c7143419543c9a10566a5fe6
+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran4.tar.gz/md5/84e64d74834762a018fa403c3bbe0f77
+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran4.tar.gz/sha512/112c073fa91a6a5e01092b0eba062ac9201e3ebbe875a98e5c214f45f05eb7ddc00939d3dd55954b136b6fbe532b7a8aa2dfa91b278d3202ebfc7d9a966fa37d
+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran5.tar.gz/md5/020c74f778e0729d5dc1267f7c6a3fd4
+OpenBLAS.v0.3.13+4.x86_64-linux-gnu-libgfortran5.tar.gz/sha512/1587083156596a8839972516fc96484dd58dbd5bd134ea3054a843f995f69c8a0a08d174ff2a6ec37961d825d5cc55f99e1daa7828587ee68669026e655c16f8
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran3.tar.gz/md5/e179ad17125cd6726c641d30e3633edd
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran3.tar.gz/sha512/139f92f57d5c56970f741421d614b2c0c8f2fced0c19ac5615cdd8f84b8c3fda96a56f938a64ccc1431d1938c15ce86fd19b5465b79e814f39e8afdf624068b4
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran4.tar.gz/md5/a19460cdfb1ab3e334c4a23a7fa40aa8
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran4.tar.gz/sha512/702ded70c3195ad624b0e03f9bdadc9be8cc5e991b26bb2d863458a908e17d27bcfcee71eec2547e4e43c8afa204103ff6fe2eefe07db4f031d757f4367d9755
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran5.tar.gz/md5/eba567bf765f5e438e8fee579c1f91b5
+OpenBLAS.v0.3.13+4.x86_64-linux-musl-libgfortran5.tar.gz/sha512/8e02ac4b56b2634e6c9481bba7ca8878ae1dd9f0c153975f7983e6aeab5cb3a03409ee3c237e066bef2da41b8f36483c3a5026b6acc85a77071e73c342cec4bc
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran3.tar.gz/md5/321c44260b7d2443a177a9839bd5a7a6
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran3.tar.gz/sha512/ac2f99d265bc44d61d2d7422afc790f9e52d8309e07c810fabab6951dd50993652f2271f591677bfe0d9ba123e7375382b59ce477b8a05d916efa093eb83cdda
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran4.tar.gz/md5/9e650bc22e9b14320396c5621e25960d
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran4.tar.gz/sha512/847a6d689e81b7296ab08b4ff9881ce1a26af39b76199189b31d10fc0a18a24b9ffafb47c3d81c285a78691fba3c823dbf4a9aa5940b65331a10cc2eaff1b7d7
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran5.tar.gz/md5/65e2f7649e70277f90165a97eb19a67d
+OpenBLAS.v0.3.13+4.x86_64-unknown-freebsd-libgfortran5.tar.gz/sha512/62602e42c89805ab3963dc2989545a8a0f17ace7193bab527b54c2ca4c5066eefc0f6ef110c0a5e43fd4d6c9e7922be7e40a1ac9813b7fd1b3873756d9785675
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran3.tar.gz/md5/ad7ad7a00aae93c8347171c2319895ed
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran3.tar.gz/sha512/a87c5e5c7917885916cd38379944f9eb4830748d5488037834cfd6f87e50b35133a94906c56fd4f9c3716e8c3ea4f00ae214b3edbad86f0a4ccf89653eba5c7b
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran4.tar.gz/md5/ab072184c8caf97ad243937701bfbeb8
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran4.tar.gz/sha512/0ed921eda9756e262a840bdc4ec4f91c419f18efc15511b3ab5eb4407bade5b73cfdc8b9e8c2d42c1cd500a98729d452040bd41fe3080d5fdec30e12841fe975
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran5.tar.gz/md5/532b261f05ee1c0f131a55ba138df9c6
+OpenBLAS.v0.3.13+4.x86_64-w64-mingw32-libgfortran5.tar.gz/sha512/407279412cc912cf56cdafd5569eddfdca92dc01a045fb3450b9cedb9c9d2efbcf01d44bf8de94c197c4bad398e911ad651ac08a3ccd3741c1ad30661451471a
diff --git a/deps/checksums/pcre b/deps/checksums/pcre
index 05a06f9844ddf..f7e1fa0c1a3ba 100644
--- a/deps/checksums/pcre
+++ b/deps/checksums/pcre
@@ -1,3 +1,5 @@
+pcre2-10.36.tar.bz2/md5/bd7e7421ff3fa2e2d5429229ecfad095
+pcre2-10.36.tar.bz2/sha512/fc2a920562c80c3d31cedd94028fab55314ae0fb168cac7178f286c344a11fc514939edc3b83b8e0b57c872db4e595fd5530fd1d4b8c779be629553e9ec965a3
 PCRE2.v10.36.0+2.aarch64-apple-darwin.tar.gz/md5/12ac3bee39df3a79f868f6463964953b
 PCRE2.v10.36.0+2.aarch64-apple-darwin.tar.gz/sha512/a1a1312931deb7f742f80886188babcf9c179ed3f156626fb23d92633fde896d1ee9b2d72cd99ae4a1f8048971b6d939e9b0b10c455d4eeec24b265968593486
 PCRE2.v10.36.0+2.aarch64-linux-gnu.tar.gz/md5/32240ccddee3040aeedcbe69ea52fcad
@@ -30,5 +32,3 @@ PCRE2.v10.36.0+2.x86_64-unknown-freebsd.tar.gz/md5/97410029c0b6ed5f7fb0d14e1f121
 PCRE2.v10.36.0+2.x86_64-unknown-freebsd.tar.gz/sha512/229e910759da2959ddef83ca89e05a050c266b8e755c85dfce6a786658be541911c3b78a0fca7dfdee1b41fbbdccf57da75cf9fe45fd2821dba8d2aaeabfd538
 PCRE2.v10.36.0+2.x86_64-w64-mingw32.tar.gz/md5/39827564bca329768e0380bd79b869fe
 PCRE2.v10.36.0+2.x86_64-w64-mingw32.tar.gz/sha512/4579049b99fca3334d726b0ca1f07524d1643a758e375b5b02b8f294ba7d9c2a4130da1a1523de29033233a8848105b3cb660e15bb4a759593405d805ee99883
-pcre2-10.36.tar.bz2/md5/bd7e7421ff3fa2e2d5429229ecfad095
-pcre2-10.36.tar.bz2/sha512/fc2a920562c80c3d31cedd94028fab55314ae0fb168cac7178f286c344a11fc514939edc3b83b8e0b57c872db4e595fd5530fd1d4b8c779be629553e9ec965a3
diff --git a/deps/checksums/suitesparse b/deps/checksums/suitesparse
index 98bf43c0c3a92..82e4a00578de6 100644
--- a/deps/checksums/suitesparse
+++ b/deps/checksums/suitesparse
@@ -25,6 +25,10 @@ SuiteSparse.v5.8.1+0.powerpc64le-linux-gnu.tar.gz/sha512/412611fd82923e042001774
 SuiteSparse.v5.8.1+0.x86_64-apple-darwin.tar.gz/md5/d266370cc574c8ca8b2b8a94d6b877f4
 SuiteSparse.v5.8.1+0.x86_64-apple-darwin.tar.gz/sha512/2377d662f91d01229e3c44ec1c5624b6361e823ff90d625acc3bab8f68ada1b3b7b67f4e56c8896f890e5780bd2fcb494b5a1f91c9c7803715eac2a2a4bb938f
 SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/md5/2545d16665a42f36fbbad9198a9668e5
+SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/md5/2545d16665a42f36fbbad9198a9668e5
+SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/md5/2545d16665a42f36fbbad9198a9668e5
+SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/sha512/66cf497a002b566a681531873da0b5dc3b4f79414068b7ae53845325afb2f56a55069ff5ba5f47de0eb961af4df18cf8495019150f163213d32aa5b12c511ce3
+SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/sha512/66cf497a002b566a681531873da0b5dc3b4f79414068b7ae53845325afb2f56a55069ff5ba5f47de0eb961af4df18cf8495019150f163213d32aa5b12c511ce3
 SuiteSparse.v5.8.1+0.x86_64-linux-gnu.tar.gz/sha512/66cf497a002b566a681531873da0b5dc3b4f79414068b7ae53845325afb2f56a55069ff5ba5f47de0eb961af4df18cf8495019150f163213d32aa5b12c511ce3
 SuiteSparse.v5.8.1+0.x86_64-linux-musl.tar.gz/md5/8a8ade173d751514e03a6987aee56ca1
 SuiteSparse.v5.8.1+0.x86_64-linux-musl.tar.gz/sha512/7393063fd2f88bf5dbe1378d6d34206bde446f1ed971d2dd334c3cca96075334f4f8ca69f7d406d532a631fd8a76d739df27dab7928e596a1d38ba9c96fbc9d6
diff --git a/stdlib/LinearAlgebra/docs/src/index.md b/stdlib/LinearAlgebra/docs/src/index.md
index 48d44d98f7ca0..e71ac571d0b9b 100644
--- a/stdlib/LinearAlgebra/docs/src/index.md
+++ b/stdlib/LinearAlgebra/docs/src/index.md
@@ -31,20 +31,20 @@ julia> inv(A)
 As well as other useful operations, such as finding eigenvalues or eigenvectors:
 
 ```jldoctest
-julia> A = [-4. -17.; 2. 2.]
+julia> A = [-4. -9.; 2. 2.]
 2×2 Matrix{Float64}:
- -4.0  -17.0
-  2.0    2.0
+ -4.0  -9.0
+  2.0   2.0
 
 julia> eigvals(A)
 2-element Vector{ComplexF64}:
- -1.0 - 5.0im
- -1.0 + 5.0im
+ -1.0 - 3.0im
+ -1.0 + 3.0im
 
 julia> eigvecs(A)
 2×2 Matrix{ComplexF64}:
-  0.945905-0.0im        0.945905+0.0im
- -0.166924+0.278207im  -0.166924-0.278207im
+  0.904534-0.0im        0.904534+0.0im
+ -0.301511+0.301511im  -0.301511-0.301511im
 ```
 
 In addition, Julia provides many [factorizations](@ref man-linalg-factorizations) which can be used to
diff --git a/stdlib/LinearAlgebra/src/symmetriceigen.jl b/stdlib/LinearAlgebra/src/symmetriceigen.jl
index 0f18d290b4d62..e98f46d13dcd1 100644
--- a/stdlib/LinearAlgebra/src/symmetriceigen.jl
+++ b/stdlib/LinearAlgebra/src/symmetriceigen.jl
@@ -99,7 +99,7 @@ julia> eigvals(A, 2:2)
 
 julia> eigvals(A)
 3-element Vector{Float64}:
- -2.1400549446402604
+ -2.140054944640259
   1.0000000000000002
   5.140054944640259
 ```
@@ -139,7 +139,7 @@ julia> eigvals(A, -1, 2)
 
 julia> eigvals(A)
 3-element Vector{Float64}:
- -2.1400549446402604
+ -2.140054944640259
   1.0000000000000002
   5.140054944640259
 ```
diff --git a/stdlib/LinearAlgebra/src/tridiag.jl b/stdlib/LinearAlgebra/src/tridiag.jl
index 8420750f8f4a1..b19e2b5de1efb 100644
--- a/stdlib/LinearAlgebra/src/tridiag.jl
+++ b/stdlib/LinearAlgebra/src/tridiag.jl
@@ -330,15 +330,15 @@ julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
 
 julia> eigvals(A)
 3-element Vector{Float64}:
- -2.1400549446402604
+ -2.140054944640259
   1.0000000000000002
   5.140054944640259
 
 julia> eigvecs(A)
 3×3 Matrix{Float64}:
-  0.418304  -0.83205      0.364299
- -0.656749  -7.39009e-16  0.754109
-  0.627457   0.5547       0.546448
+  0.418304   0.83205      0.364299
+ -0.656749   8.21121e-16  0.754109
+  0.627457  -0.5547       0.546448
 
 julia> eigvecs(A, [1.])
 3×1 Matrix{Float64}:
diff --git a/stdlib/LinearAlgebra/test/schur.jl b/stdlib/LinearAlgebra/test/schur.jl
index feb0ef8513b89..5d2589dd44afe 100644
--- a/stdlib/LinearAlgebra/test/schur.jl
+++ b/stdlib/LinearAlgebra/test/schur.jl
@@ -134,4 +134,30 @@ aimg  = randn(n,n)/2
     end
 end
 
+@testset "Generalized Schur convergence" begin
+    # Check for convergence issues, #40279
+    problematic_pencils = [
+        (   ComplexF64[0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0; 3.7796350217469814 -3.3125635598133054 0.0 0.0 0.0 0.0 0.0 0.0 6.418270043493963 -6.625127119626611 0.0 0.0 0.0 0.0 0.0 -1.0; -3.312563559813306 3.779635021746982 0.0 0.0 0.0 0.0 0.0 0.0 -6.625127119626612 6.418270043493964 -1.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 3.7796350217469814 0.0 0.0 -3.3125635598133054 0.0 0.0 0.0 -1.0 6.418270043493963 0.0 0.0 -6.625127119626611 0.0 0.0; 0.0 0.0 0.0 3.779635021746982 -3.312563559813306 0.0 0.0 0.0 0.0 0.0 0.0 6.418270043493964 -6.625127119626612 0.0 -1.0 0.0; 0.0 0.0 0.0 -3.3125635598133054 3.7796350217469814 0.0 0.0 0.0 0.0 0.0 0.0 -6.625127119626611 6.418270043493963 -1.0 0.0 0.0; 0.0 0.0 -3.312563559813306 0.0 0.0 3.779635021746982 0.0 0.0 0.0 0.0 -6.625127119626612 0.0 -1.0 6.418270043493964 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 3.7796350217469814 -3.3125635598133054 0.0 0.0 0.0 -1.0 0.0 0.0 6.418270043493963 -6.625127119626611; 0.0 0.0 0.0 0.0 0.0 0.0 -3.312563559813306 3.779635021746982 -1.0 0.0 0.0 0.0 0.0 0.0 -6.625127119626612 6.418270043493964],
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+        )]
+
+    for (A, B) in problematic_pencils
+        f = schur(A, B)
+        @test f.Q*f.S*f.Z' ≈ A
+        @test f.Q*f.T*f.Z' ≈ B
+    end
+end
+
 end # module TestSchur
diff --git a/stdlib/OpenBLAS_jll/Project.toml b/stdlib/OpenBLAS_jll/Project.toml
index e732272c12e09..b3bf88b2afb57 100644
--- a/stdlib/OpenBLAS_jll/Project.toml
+++ b/stdlib/OpenBLAS_jll/Project.toml
@@ -1,6 +1,6 @@
 name = "OpenBLAS_jll"
 uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
-version = "0.3.13+3"
+version = "0.3.13+4"
 
 [deps]
 CompilerSupportLibraries_jll = "e66e0078-7015-5450-92f7-15fbd957f2ae"